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Tomás Chacón Rebollo

Publications and source records attributed to Tomás Chacón Rebollo.

16 recordsLinked to original sources
arXiv2025

Reduced-Order Time Splitting for Navier-Stokes with Open Boundaries

In this work, we propose a Proper Orthogonal Decomposition-Reduced Order Model (POD-ROM) applied to time-splitting schemes for solving the Navier-Stokes equations with open boundary conditions. In this method, we combine three strategies to reduce the computing time to solve NSE: time splitting, reduction of the computational domain through non-standard treatment of open boundary conditions and reduced order modelling. To make the work self-contained, we first present the formulation of the time-splitting scheme applied to the Navier-Stokes equations with open boundary conditions, employing a first-order Euler time discretization and deriving the non-standard boundary condition for pressure. Then, we construct a Galerkin projection-based ROM using POD with two different treatments of the pressure boundary condition on the outlet. We propose a comparative performance analysis between the standard projection-based POD-ROM (fully intrusive) and a hybrid POD-ROM that combines a projection-based approach (intrusive) with a data-driven technique (non-intrusive) using Radial Basis Functions (RBF). We illustrate this comparison through two different numerical tests: the flow in a bifurcated tube and the benchmark numerical test of the flow past cylinder, numerically investigating the efficiency and accuracy of both ROMs.

math.NA↗
arXiv2025

Variational Multiscale Evolve and Filter Strategies for Convection-Dominated Flows

The evolve-filter (EF) model is a filter-based numerical stabilization for under-resolved convection-dominated flows. EF is a simple, modular, and effective strategy for both full-order models (FOMs) and reduced-order models (ROMs). It is well-known, however, that when the filter radius is too large, EF can be overdiffusive and yield inaccurate results. To alleviate this, EF is usually supplemented with a relaxation step. The relaxation parameter, however, is very sensitive with respect to the model parameters. In this paper, we propose a novel strategy to alleviate the EF overdiffusivity for a large filter radius. Specifically, we leverage the variational multiscale (VMS) framework to separate the large resolved scales from the small resolved scales in the evolved velocity, and we use the filtered small scales to correct the large scales. Furthermore, in the new VMS-EF strategy, we use two different ways to decompose the evolved velocity: the VMS Evolve-Filter-Filter-Correct (VMS-EFFC) and the VMS Evolve-Postprocess-Filter-Correct (VMS-EPFC) algorithms. The new VMS-based algorithms yield significantly more accurate results than the standard EF in both the FOM and the ROM simulations of a flow past a cylinder at Reynolds number Re = 1000.

math.NA↗
arXiv2023

On a certified VMS-Smagorinsky Reduced Basis model with LPS pressure stabilisation

In this work we introduce a certified Reduced Basis VMS-Smagorinsky turbulence model with local projection stabilisation (LPS) on the pressure. We prove its stability for Taylor-Hood discretisations of velocity-pressure. We construct an \textit{a posteriori} error estimator for the snapshot selection through a Greedy algorithm, based on the Brezzi-Rappaz-Raviart theory of approximation of non-singular branches of non-linear PDEs. The Empirical Interpolation Method (EIM) is used for the approximation of the non-linear terms. We present some numerical tests in which we show an improved speedup on the computation of the reduced basis problem with the LPS pressure stabilisation, with respect to the method of using pressure supremizers.

math.NA↗
arXiv2023

A boundary-oriented reduced Schwarz domain decomposition technique for parametric advection-diffusion problems

We present in this paper the results of a research motivated by the need of a very fast solution of thermal flow in solar receivers. These receivers are composed by a large number of parallel pipes with the same geometry. We have introduced a reduced Schwarz algorithm that skips the computation in a large part of the pipes. The computation of the temperature in the skep domain is replaced by a reduced mapping that provides the transmission conditions. This reduced mapping is computed in an off-line stage. We have performed an error analysis of the reduced Schwarz algorithm, proving that the error is bounded in terms of the linearly decreasing error of the standard Schwarz algorithm, plus the error stemming from the reduction of the trace mapping. The last error is asymptotically dominant in the Schwarz iterative process. We obtain $L^2$ errors below $2\%$ with relatively small overlapping lengths.

math.NA↗
arXiv2023

A Reduced Basis LES turbulence model based upon Kolmogorov's equilibrium turbulence theory

In this work, we introduce an a posteriori error indicator for the reduced basis modelling of turbulent flows. It is based upon the $k^{-5/3}$ Kolmogorov turbulence theory, thus it may be applied to any numerical discretisation of LES turbulence models. The main idea of this indicator is that if the full-order solution and the Reduced Order solution are close enough, then their flow energy spectrum within the inertial range should also be close. We present some numerical tests which supports that the use of this indicator is helpful, obtaining large computational speed-ups. We use as full-order model a Finite Element discretisation of the unsteady LES Smagorinsky turbulence model.

physics.flu-dyn↗
arXiv2023

Uniform Bounds with Difference Quotients for Proper Orthogonal Decomposition Reduced Order Models of the Burgers Equation

In this paper, we prove uniform error bounds for proper orthogonal decomposition (POD) reduced order modeling (ROM) of Burgers equation, considering difference quotients (DQs), introduced in [26]. In particular, we study the behavior of the DQ ROM error bounds by considering $L^2(Ω)$ and $H^1_0(Ω)$ POD spaces and $l^{\infty}(L^2)$ and natural-norm errors. We present some meaningful numerical tests checking the behavior of error bounds. Based on our numerical results, DQ ROM errors are several orders of magnitude smaller than noDQ ones (in which the POD is constructed in a standard way, i.e., without the DQ approach) in terms of the energy kept by the ROM basis. Further, noDQ ROM errors have an optimal behavior, while DQ ROM errors, where the DQ is added to the POD process, demonstrate an optimality/super-optimality behavior. It is conjectured that this possibly occurs because the DQ inner products allow the time dependency in the ROM spaces to make an impact.

math.NA↗
arXiv2022

Least-squares driven stabilised finite element solution of advection-dominated flow problems

In this article, we address the solution of advection-dominated flow problems by stabilised methods, by means of least-squares computed stabilised coefficients. As main methodological tool, we introduce a data-driven off-line/on-line strategy to compute them with low computational cost. We compare the errors provided by the least-squares stabilised coefficients to those provided by several previously established stabilised coefficients within the solution of advection-diffusion and Navier-Stokes flows, on structured and un-structured grids, with and Lagrange Finite Elements up to third degree of interpolation. In all tested flows the least-squares stabilised coefficients provide quasi-optimal errors. We conclude that the least-squares procedure is a rewarding procedure, worth to be applied to general stabilised solutions of general flow problems.

math.NA↗
arXiv2022

Error analysis of a residual-based stabilization-motivated POD-ROM for incompressible flows

This article presents error bounds for a velocity-pressure segregated POD reduced order model discretization of the Navier-Stokes equations. The stability is proven in L infinity (L 2 ) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L 2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence.

math.NA↗
arXiv2022

Spectral Variational Multi-Scale method for parabolic problems. Application to 1D transient advection-diffusion equations

In this work, we introduce a Variational Multi-Scale (VMS) method for the numerical approximation of parabolic problems, where sub-grid scales are approximated from the eigenpairs of associated elliptic operator. The abstract method is particularized to the one-dimensional advection-diffusion equations, for which the sub-grid components are exactly calculated in terms of a spectral expansion when the advection velocity is approximated by piecewise constant velocities on the grid elements. We prove error estimates that in particular imply that when Lagrange finite element discretisations in space are used, the spectral VMS method coincides with the exact solution of the implicit Euler semi-discretisation of the advection-diffusion problem at the Lagrange interpolation nodes. We also build a feasible method to solve the evolutive advection-diffusion problems by means of an offline/online strategy with reduced computational complexity. We perform some numerical tests in good agreement with the theoretical expectations, that show an improved accuracy with respect to several stabilised methods.

math.NA↗
arXiv2020

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that let us obtain an affine decomposition with respect to the parameters. We construct an a posterior error estimator, based upon the Brezzi-Rappaz-Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

math.NA↗
arXiv2019

A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations

In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framework and applied to advection-diffusion-reaction problems. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.

physics.comp-ph↗
arXiv2018

VMS spectral solution of two-dimensional advection-diffusion problems with anisotropic velocity

In this article, we extend the Variational Multi-scale method with spectral approximation of the sub-scales to two-dimensional advection-diffusion problems. The spectral VMS method is cast for low-order elements as a standard VMS method with specific stabilized coefficients associated to a component of the advection velocity. We compute the stabilized coefficients for grids of isosceles right triangles and right quadrilaterals, based upon the explicit computation of the eigen-pairs of the advection-diffusion operator with Dirichlet boundary conditions. To reduce the computing time, the stabilized coefficients are computed at the nodes of a grid in an off-line step, and then interpolated by a fast procedure in the on-line computation. Finally, we present some numerical tests, first with constant velocity and after that with anisotropic variable velocity, in order to compare our results with those provided by other stabilization coefficients. We observe a relevant accuracy gain for moderately large grid Péclet numbers for variable advection velocity.

math.NA↗
arXiv2018

Variational multi-scale spectral solution of convection-dominated parabolic problems

In this work, we consider an extension to parabolic problems of the variational multiscale method with spectral approximation of the sub-scales. We first discretize in time using a finite difference scheme and second, apply the generalization of the spectral variational multi-scale method. To obtain error estimations in convection-dominated flows, we find a helpful link between the stabilized term expressed in terms of Green's functions and in terms of spectral functions. Finally, we present some numerical tests to show the reliability of the method. We consider the stationary one-dimensional advection-diffusion-reaction equation and the evolutive one-dimensional advection-diffusion equation.

math.NA↗
arXiv2017

Streamline derivative projection-based POD-ROM for convection-dominated flows. Part I : Numerical Analysis

We introduce improved Reduced Order Models (ROM) for convection-dominated flows. These non-linear closure models are inspired from successful numerical stabilization techniques used in Large Eddy Simulations (LES), such as Local Projection Stabilization (LPS), applied to standard models created by Proper Orthogonal Decomposition (POD) of flows with Galerkin projection. The numerical analysis of the fully Navier-Stokes discretization for the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Also, we suggest an efficient practical implementation of the stabilization term, where the stabilization parameter is approximated by the Discrete Empirical Interpolation Method (DEIM).

math.NA↗
arXiv2017

On a certified Smagorinsky reduced basis turbulence model

In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term. We present some numerical tests, programmed in FreeFem++, in which we show an speedup on the computation by factor larger than 1000 in benchmark 2D flows.

math.NA↗