arXiv subjects

Ignacio Romero

Publications and source records attributed to Ignacio Romero.

At least 19 recordsLinked to original sources

arXiv2026

A model and a finite element approximation of the mixed-dimensionality diffusion problem

We present the formulation of a boundary value problem that models the coupled behavior of a three-dimensional diffusive solid with one-dimensional diffusive fibers embedded inside it. We introduce a variational statement of the problem that identifies the linked diffusive fields as energy minimizers under a coupling constraint. This saddle-point problem is proved to be well posed. Then, we introduce a finite element discretization of the proposed boundary value problem, and we prove the convergence of the finite element solution to the exact one. The most significant feature of this approximation is that the meshes of the bodies need not be conforming. Numerical examples confirm the theoretical results

math.NA↗

arXiv2026

Identification of optimal history variables and corresponding hereditary laws in linear viscoelasticity

We develop an operator-theoretic formulation of hereditary constitutive models and characterize optimal finite-rank internal-variable approximations in the sense of Kolmogorov $N$-widths. The history operator is shown to be compact under natural assumptions on the relaxation kernel, thereby admitting optimal low-rank approximations. The resulting reduced models inherit thermodynamic consistency, stability, and provable approximation bounds. An analysis clarifies the structural relation between hereditary representations and internal-variable theories and provides a rigorous basis for reduced-order modeling in computational mechanics. Selected numerical examples showcase optimal convergence of approximations with respect to rank and sampling.

math-ph↗

arXiv2026

On the pure traction problem of linear elasticity: a regularized formulation and its robust approximation

The pure traction problem of elasticity appears frequently in engineering applications, and its complexity stems from the fact that its solution is unique only up to (infinitesimal) rigid body motions. When finite elements are employed to approximate this problem, one solution is typically singled out by applying carefully selected boundary conditions on the discrete model or by imposing global constraints on the deformation. However, neither of these strategies is both simple and computationally efficient. In this work, we propose a new approach to solving the pure traction problem that overcomes existing limitations. Our method builds on a regularized form of the problem whose solution is shown to be unique, converges to the original solution of minimal norm, and can be approximated with finite elements in a straightforward way, without additional degrees of freedom. Additionally, we analyze the situation in which the approximation of the solution domain renders the loading of the discretized problem non-equilibrated, making the problem ill-posed. In this case, we propose a regularized predictor--corrector finite element formulation that handles the incompatibilities of the loading, providing a solution that converges to that of the original Neumann problem as the mesh size and the regularizing parameter tend to zero. Numerical examples illustrate the effectiveness of the proposed approach for representative problems in mechanics where pure traction boundary conditions appear.

math.NA↗

arXiv2026

A Framework for the Bayesian Calibration of Complex and Data-Scarce Models in Applied Sciences

In this work, we review the theory involved in the Bayesian calibration of complex computer models, with particular emphasis on their use for applications involving computationally expensive simulations and scarce experimental data. In the article, we present a unified framework that incorporates various Bayesian calibration methods, including well-established approaches. Furthermore, we describe their implementation and use with a new, open-source Python library, ACBICI (A Configurable BayesIan Calibration and Inference Package). All algorithms are implemented with an object-oriented structure designed to be both easy to use and readily extensible. In particular, single-output and multiple-output calibration are addressed in a consistent manner. The article completes the theory and its implementation with practical recommendations for calibrating the problems of interest. These guidelines -- currently unavailable in a unified form elsewhere -- together with the open-source Python library, are intended to support the reliable calibration of computational codes and models commonly used in engineering and related fields. Overall, this work aims to serve both as a comprehensive review of the statistical foundations and (computational) tools required to perform such calculations, and as a practical guide to Bayesian calibration with modern software tools.

stat.CO↗

arXiv2025

Bayesian Calibration and Model Assessment of Cell Migration Dynamics with Surrogate Model Integration

Computational models provide crucial insights into complex biological processes such as cancer evolution, but their mechanistic nature often makes them nonlinear and parameter-rich, complicating calibration. We systematically evaluate parameter probability distributions in cell migration models using Bayesian calibration across four complementary strategies: parametric and surrogate models, each with and without explicit model discrepancy. This approach enables joint analysis of parameter uncertainty, predictive performance, and interpretability. Applied to a real data experiment of glioblastoma progression in microfluidic devices, surrogate models achieve higher computational efficiency and predictive accuracy, whereas parametric models yield more reliable parameter estimates due to their mechanistic grounding. Incorporating model discrepancy exposes structural limitations, clarifying where model refinement is necessary. Together, these comparisons offer practical guidance for calibrating and improving computational models of complex biological systems.

math.AP↗

arXiv2025

Embedding structures in continua: linear models and finite element discretizations

This work describes models and numerical approximations that describe the mechanical behavior of deformable continua with embedded structural members, such as rigid bodies, beams, shells, etc. The continuum formulation extends an idea first presented in the context of the Arlequin method and constrains the kinematics of the two types of bodies to be compatible in the energy sense. In the article, we exploit the shared similarities of all structural theories to introduce a general framework for energetically coupling the latter with continua. In addition, we show that the problems, as well as their finite element approximations, are well-posed. Numerical examples of bodies with inclusions, fibers, and embedded surfaces are provided to illustrate the generality and robustness of the approach.

math.NA↗

arXiv2025

A fully variational numerical method for structural topology optimization based on a Cahn-Hilliard model

We formulate a novel numerical method suitable for the solution of topology optimization problems in solid mechanics. The most salient feature of the new approach is that the space and time discrete equations of the numerical method can be obtained as the optimality conditions of a single incremental potential. The governing equations define a gradient flow of the mass in the domain that maximizes the stiffness of the proposed solid, while exactly preserving the mass of the allocated material. Moreover, we propose a change of variables in the model equations that constrains the value of the density within admissible bounds and a continuation strategy that speeds up the evolution of the flow. The proposed strategy results in a robust and efficient topology optimization method that is exactly mass-preserving, does not employ Lagrange multipliers, and is fully variational.

math.NA↗

arXiv2025

An optimal-transport finite-particle method for driven mass diffusion

We formulate a finite-particle method of mass transport that accounts for general mixed boundary conditions. The particle method couples a geometrically-exact treatment of advection; Wasserstein gradient-flow dynamics; and a Kullback-Leibler representation of the entropy. General boundary conditions are enforced by introducing an adsorption/depletion layer at the boundary wherein particles are added or removed as dictated by the boundary conditions. We demonstrate the range and scope of the method through a number of examples of application, including absorption of particles into a sphere and flow through pipes of square and circular cross section, with and without occlusions. In all cases, the solution is observed to converge weakly, or in the sense of local averages.

math.NA↗

arXiv2024

A variational method for the simulation of hydrogen diffusion in metals

We present a new method for the approximate solution of the strongly coupled, nonlinear stress-diffusion problem that appears when modeling hydrogen transport in metals. The most salient feature of the proposed approximation is that it is fully variational, meaning that all the discrete equations are obtained from the optimality conditions of an incremental potential, even for inelastic mechanical behavior. Like other variational methods, the proposed algorithm has remarkable properties, including the symmetry of the tangent operator, making its solution extremely efficient compared to other similar methods available in the literature.

cond-mat.mtrl-sci↗

arXiv2024

Model-Based Reinforcement Learning Control of Reaction-Diffusion Problems

Mathematical and computational tools have proven to be reliable in decision-making processes. In recent times, in particular, machine learning-based methods are becoming increasingly popular as advanced support tools. When dealing with control problems, reinforcement learning has been applied to decision-making in several applications, most notably in games. The success of these methods in finding solutions to complex problems motivates the exploration of new areas where they can be employed to overcome current difficulties. In this paper, we explore the use of automatic control strategies to initial boundary value problems in thermal and disease transport. Specifically, in this work, we adapt an existing reinforcement learning algorithm using a stochastic policy gradient method and we introduce two novel reward functions to drive the flow of the transported field. The new model-based framework exploits the interactions between a reaction-diffusion model and the modified agent. The results show that certain controls can be implemented successfully in these applications, although model simplifications had to be assumed.

math.OC↗

arXiv2023

The energy-stepping Monte Carlo method: an exactly symmetry-preserving, a Hamiltonian Monte Carlo method with a 100% acceptance ratio

We introduce the energy-stepping Monte Carlo (ESMC) method, a Markov chain Monte Carlo (MCMC) algorithm based on the conventional dynamical interpretation of the proposal stage but employing an energy-stepping integrator. The energy-stepping integrator is quasi-explicit, symplectic, energy-conserving, and symmetry-preserving. As a result of the exact energy conservation of energy-stepping integrators, ESMC has a 100\%\ acceptance ratio of the proposal states. Numerical tests provide empirical evidence that ESMC affords a number of additional benefits: the Markov chains it generates have weak autocorrelation, it has the ability to explore distant characteristic sets of the sampled probability distribution and it yields smaller errors than chains sampled with Hamiltonian Monte Carlo (HMC) and similar step sizes. Finally, ESMC benefits from the exact symmetry conservation properties of the energy-stepping integrator when sampling from potentials with built-in symmetries, whether explicitly known or not.

math-ph↗

arXiv2022

Linking discrete and continuum diffusion models: Well-posedness and stable finite element discretizations

In the context of mathematical modeling, it is sometimes convenient to integrate models of different nature. These types of combinations, however, might entail difficulties even when individual models are well-understood, particularly in relation to the well-posedness of the ensemble. In this article, we focus on combining two classes of dissimilar diffusive models: the first one defined over a continuum and the second one based on discrete equations that connect average values of the solution over disjoint subdomains. For stationary problems, we show unconditional stability of the linked problems and then the stability and convergence of its discretized counterpart when mixed finite elements are used to approximate the model on the continuum. The theoretical results are highlighted with numerical examples illustrating the effects of linking diffusive models. As a side result, we show that the methods introduced in this article can be used to infer the solution of diffusive problems with incomplete data.

math.NA↗

arXiv2022

Analysis and design of bistable and thermally reversible metamaterials inspired by shape-memory alloys

In this work, we study lattice structures that exhibit a bistable behavior, i. e., they can snap from one stable state to another, and are also completely reversible, capable of reverting back to its original state through a heat treatment. We design this behavior by constructing lattice structures using networks of nonlinear springs that display tension-compression asymmetry and have different thermal expansion coefficients. The mismatch in the thermal expansion coefficients induces residual stresses in the springs which results in the lattice structure exhibiting bistability at low temperatures and monostability at high temperatures. This behavior mimics the crystallographic phase transformations of shape memory alloys, but here artificially introduced in a structural lattice. By analyzing a representative unit cell, we quantify the effect that the stiffness and the thermal expansion coefficient of the springs have on the stability of the structural lattice. In addition, for simple 2D lattices, using the concept of universal unfoldings of singularity theory, we perform a perturbation analysis to identify the key variables of the structure where controlling defects is important, as they lead to drastic changes in the bifurcation behavior of the lattice. Finally, we verify numerically our analytical predictions in both 2D and 3D simulations using continuation techniques. The examples proposed confirm that the bistable and reversible features of the unit cell carry on to the macroscale, opening the route for the design of lattice structures for energy absorption applications that can hea} with a heat treatment.

physics.class-ph↗

arXiv2020

Energy-momentum conserving integration schemes for molecular dynamics

We address the formulation and analysis of energy and momentum conserving time integration schemes in the context of particle dynamics, and in particular atomic systems. The article identifies three critical aspects of these models that demand a careful analysis when discretized: first, the treatment of periodic boundary conditions; second, the formulation of approximations of systems with three-body interaction forces; third, their extension to atomic systems with functional potentials. These issues, and in particular their interplay with Energy-Momentum integrators, are studied in detail. Novel expressions for these time integration schemes are proposed and numerical examples are given to illustrate their performance.

physics.comp-ph↗

arXiv2019

The rotating rigid body model based on a non-twisting frame

This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange-D'Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy-momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations.

physics.class-ph↗

arXiv2019

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

math.NA↗

arXiv2019

Connecting beams and continua: variational basis and mathematical analysis

We present a new variational principle for linking models of beams and deformable solids, providing also its mathematical analysis. Despite the apparent differences between the two types of governing equations, it will be shown that the equilibrium of systems combining beams and solids can be obtained from a joint constrained variational principle and that the resulting boundary-value problem is well posed.

math.NA↗

arXiv2019

Structural models based on 3D constitutive laws: variational structure and numerical solution

In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For other, more complex and interesting situations, kinematic and kinetic hypotheses need to be invoked, and a constrained three-dimensional constitutive relation has to be employed at every point of the section in order to implement non-linear and dissipative constitutive laws into dimensionally reduced structural models. In this article we explain in which sense reduced constitutive models can be expressed as minimization problems, helping to formulate the global equilibrium as a single optimization problem. Casting the problem this way has implications from the mathematical and numerical points of view, naturally defining error indicators. General purpose solution algorithms for constrained material response, with and without optimization character, are discussed and provided in an open-source library.

physics.class-ph↗