arXiv subjects

Ramon Codina

Publications and source records attributed to Ramon Codina.

9 recordsLinked to original sources

arXiv2026

Existence of thermodynamically consistent solutions for data-driven porous media problems

Data-Driven Computational Mechanics (DDCM) replaces traditional phenomenological constitutive models by directly reformulating boundary-value problems in terms of local material state data obtained from experiments or fine-scale simulations. Standard DDCM formulations, only enforcing equilibrium and compatibility, do not inherently guarantee compliance with the second law of thermodynamics. This breakdown occurs particularly when input material data sets are subject to noise or local physical non-admissibility. In this work, we present a variational DDCM framework specifically tailored to diffusion--reaction problems. Taking advantage of the simplicity of the thermodynamic constraint in gradient-flux systems, we propose an augmented formulation that explicitly enforces the second law of thermodynamics as a hard constraint within the energy-minimization problem. Although the set of thermodynamically admissible states is non-convex and fails to be weakly closed in the ambient phase space, we establish existence of minimizers by proving that the intersection of the admissible set with the subspace of fields that are compatible and in equilibrium is weakly sequentially closed via a compensated compactness argument. To enable practical computations, we analyze both a Lagrange multiplier formulation and a penalization scheme. We prove the $\Gamma$-convergence of the penalized functionals to the exact constrained problem and establish a fully discrete convergence framework incorporating spatial finite-element discretization and empirical data-set approximations. Numerical experiments confirm that the proposed penalty scheme effectively restores thermodynamic consistency even in the presence of severely corrupted material data.

math.AP↗

arXiv2026

A numerical study to analyze the interplay of Weissenberg number and viscosity ratio in a log-strain tensorial model for viscoelastic fluids

We present a computational study aimed at exploring the different and independent roles of the Weissenberg number and of the ratio between the polymeric and solvent viscosity contributions in a viscoelastic fluid model. The tensorial model under investigation, recently proposed, is based on a logarithmic relation between the elastic (or recoverable) strain and the elastic stress. In this model, the elastic strain plays the role of a conformation tensor and its evolution equation inherently preserves its determinant and positive definiteness. These properties are also enforced in the computational method employed in the study. A finite-difference discretization in time is combined with a stabilized mixed finite element formulation based on the Variational Multiscale method for the spatial discretization and with a generalized Lie derivative approach for the advection terms. The behavior of the model is analyzed in paradigmatic pressure-driven flows and we find that the value of the viscosity ratio is crucial in determining to which extent non-Newtonian flow profiles are observed upon increasing the Weissenberg number. By comparing the solutions of the log-strain tensorial model with those of a suitable Generalized Newtonian Fluid model, we show that flow-type dependence plays a significant role even in the simple planar flow past a cylinder.

cond-mat.soft↗

arXiv2025

A variational multiscale approach to PDE-constrained optimization problems arising in Data-Driven Computational Mechanics

We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous setting, we establish well-posedness of such concomitant formulations. Then, we propose stable and consistent finite element approximations for these underlying primal and dual problems relying on the Variational MultiScale framework. For quasi-uniform finite element partitions, we investigate approximations' general properties and establish well-posedness for two canonical choices of the sub-grid scales, i.e., the Algebraic Sub-Grid Scale and Orthogonal Sub-Grid Scale. Moreover, for continuous finite element functions, we are able to move back and forth between the discrete primal and dual formulations only by changing the design of the stabilization parameters. To conclude, we stress-test the proposed approximations through a series of progressively sophisticated cases, providing both a comparative and qualitative assessment of their numerical performance.

math.NA↗

arXiv2025

A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept

In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family.

math.NA↗

arXiv2025

A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems

This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for error estimation is based on the variational multiscale (VMS) concept, where the solution is decomposed into resolved (finite element) and unresolved (sub-grid) scales. In this work, we propose an orthogonal sub-grid scale (OSGS) method for a goal-oriented error estimation in VMS discretizations. In the OSGS approach, the space of the sub-grid scales (SGSs) is orthogonal to the finite element space. The error is estimated in the quantity of interest, given by the linear functional $Q(u)$ of the unknown $u$. If the SGS $u'$ is estimated, the error in the quantity of interest can be approximated by $Q(u')$. Our approach is compared with a duality-based a posteriori error estimation method, which requires the solution of an additional auxiliary problem. The results indicate that both methods yield similar error estimates, whereas the VMS-based explicit approach is computationally less expensive than the duality-based implicit approach. Numerical tests demonstrated the effectiveness of our proposed error estimation techniques in terms of the quantity of interest functionals.

math.NA↗

arXiv2023

Stabilized finite elements for the solution of the Reynolds equation considering cavitation

The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection-diffusion-reaction (CDR) equation. Its solution by finite elements is prone to oscillations in convection-dominated regions, which are present whenever cavitation occurs. We propose a stabilized finite-element method that is based on the variational multiscale method and exploits the concept of orthogonal subgrid scales. We demonstrate that this approach only requires one additional term in the weak form to obtain a stable method that converges optimally when performing mesh refinement.

math.NA↗

arXiv2023

Finite element formulations for Maxwell's eigenvalue problem using continuous Lagrangian interpolations

We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell-Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method we consider consists of an augmented formulation with the introduction of a mesh dependent term that can be regarded as the Laplacian of the multiplier of the divergence constraint. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems. We investigate the numerical performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

math.NA↗

arXiv2020

Three-Field Fluid-Structure Interaction by Means of the Variational Multiscale Method

Three-field Fluid-Structure Interaction (FSI) formulations for fluid and solid are applied and compared to the standard two field-one field formulation for fluid and solid, respectively. Both formulations are applied in a non linear setting for a Newtonian fluid and a neo-Hookean solid in an updated Lagrangian form, both approximated using finite elements and stabilized by means of the Variational Multiscale (VMS) Method to permit the use of arbitrary interpolations. It is shown that this type of coupling leads to a more stable solution. Even though the new formulation poses the necessity of additional degrees of freedom, it is possible to achieve the same degree of accuracy as standard FSI by means of coarser meshes, thus making the method competitive. We enhance the stability of the formulation by assuming that the sub-grid scales of the model evolve through time. Benchmarking of the formulation is carried out. Numerical results are presented for semi-stationary and a fully transient cases for well known benchmarks for 2D and 3D scenarios.

math.NA↗

arXiv2016

A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement-pressure) and the three-field (stress-displacement-pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.

math.NA↗