Matemática Aplicada

Permanent URI for this collectionhttps://hdl.handle.net/10347/34398

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An arbitrary Lagrangian-Eulerian semi-implicit hybrid method for continuum mechanics with GLM cleaning
(Elsevier, 2026-05-01) Busto Ulloa, Saray; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
This paper proposes a novel semi-implicit arbitrary Lagrangian-Eulerian (ALE) method for the solution of the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics. To handle the curl-free-type involutions arising in the solid limit of the model, the original system is augmented by adopting a thermodynamically compatible generalized Lagrangian multiplier (GLM) approach. Next, an operator splitting strategy decouples the computation of fast pressure waves from the bulk velocity of the medium yielding a transport subsystem, containing convective terms and non-conservative products, and a Poisson-type subsystem, for the pressure. A second splitting yields an ODE subsystem comprising only the potentially stiff source terms, responsible for the relaxation of the model between its fluid and solid limits. The mesh motion can be driven by two sources: the local fluid velocity and a prescribed boundary displacement. For the spatial discretisation, we employ unstructured staggered grids, with the pressure defined on the primal mesh and all remaining variables on the dual grid. The transport subsystem is advanced via an explicit finite volume method, in which integration over closed space-time control volumes ensures verification of the geometric conservation law (GCL). On the other hand, implicit continuous finite elements are used for the discretisation of the pressure subsystem and an implicit DIRK scheme is employed to solve the ODE subsystem. Consequently, the proposed approach is well suited to address all Mach number flows. A comprehensive set of benchmarks is employed to assess the accuracy and robustness of the proposed methodology in tackling both fluid and solid mechanics problems.
Mathematical analysis of a levitation model
(Elsevier, 2025-12-23) Muñoz Sola, Rafael; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
The aim of this paper is to study a model of electromagnetic levitation for a metallic rigid body. The model is constituted by the transient linear model of eddy currents under the hypothesis of axisymmetry, written in terms of a magnetic potential vector, coupled with an ODE which governs the vertical motion of the body. The electromagnetic model is a parabolic-elliptic PDE which parabolicity region is the position occupied by the body, which changes with time. Besides, Lorentz force appears in the RHS of the ODE. Thus, the model exhibits a coupling of geometrical nature. We establish the existence and uniqueness of solution of the coupled problem and we study its maximally defined solution. In particular, we prove that a blow-up of the velocity of the body cannot happen. Our techniques involve: a reformulation of the coupled problem as a causal differential equation, an adaptation of the theory about this kind of equations and a result of locally Lipschitz dependence of the magnetic potential vector with respect to the velocity of the body.
A short note on obtaining the energy release rate for hyperelastic materials via configurational forces
(Sage Publications, 2025-12-24) Cao Rial, María Teresa; Quintela Estévez, Peregrina; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
This work aims to derive an expression for the energy release rate for hyperelastic materials using the configurational force framework. As a specific example, the classical expression of the path independent J-integral for elastic materials undergoing small deformations will be derived
An analytical 1D model for computing low-frequency electromagnetic fields in material layers: Application to metallurgical furnaces
(Elsevier, 2024-11-14) Fromreide, Mads; Gómez Pedreira, María Dolores; Halvorsen, Svenn Anton; Salgado Rodríguez, María del Pilar; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
An analytical one-dimensional model for the distribution of electric fields within multiple material layers is developed and analyzed. The model originates from the study of large three-phase electric smelting furnaces for ferroalloys and is derived from the low-frequency time-harmonic Maxwell's equations. A solution is obtained for a general case with N layers of material with different electromagnetic properties. A practical demonstration of the utility of the model is given through an application to a multilayer configuration representing the lining and casing in a FeMn furnace, with validation against 2D simulations. In addition, for a specific two-layer scenario with a highly conductive material, an approximate solution for the adjacent layer is derived. This approximation allows the distribution of the adjacent layer to depend only on its individual properties, and shows that the dissipated power reaches a maximum value when the skin depth/thickness ratio is around one. Comparative analysis between the analytical model and 2D simulations shows good qualitative agreement.
A pure-Lagrangian finite element approach for solving thermo-electrical-mechanical models. Application to electric upsetting
(Elsevier, 2025-08-28) Benítez García, Marta; Bermúdez de Castro López-Varela, Alfredo; Fontán Muíños, Pedro; Salgado Rodríguez, María del Pilar; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
In this paper, we introduce a novel numerical procedure for solving fully coupled thermo-electrical-mechanical problems using implicit Runge–Kutta time integration within a purely Lagrangian finite element framework. Our formulation, grounded in continuum mechanics, accurately captures the interdependence of mechanical, thermal, and electrical effects under large deformations. It features a fully coupled thermo-electrical-mechanical Lagrangian model with an elasto-viscoplastic constitutive law, considers six primary variables –velocity, temperature, electric potential, plastic deformation gradient, an internal strain hardening variable, and a Lagrange multiplier for enforcing contact conditions– and employs a pure-Lagrangian description. This ensures the computational domain remains fixed and known a priori, simplifies the tracking of free surfaces, and eliminates convective terms. To validate our approach, we solve several axisymmetric benchmark problems and analyze convergence rates in both time and space. Moreover, our numerical results show excellent agreement with the solution obtained using commercial packages for an in-die electric upsetting process.
Analysis of difference schemes for the Fokker–Planck angular diffusion operator
(Elsevier, 2025-03-01) López Pouso, Óscar; Segura Sala, Jose Javier; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
This paper is dedicated to the mathematical analysis of difference schemes for discretizing the angular diffusion operator present in the azimuth–independent Fokker–Planck equation. The study establishes sets of sufficient conditions to ensure that the schemes achieve convergence of order 2, and provides insights into the rationale behind certain widely recognized discrete ordinates methods. In the process, interesting properties regarding Gaussian nodes and weights, which until now have remained unnoticed by mathematicians, naturally emerge.
A comparison of lumped parameter models and Maxwell’s equations for wireless power transfer
(Springer, 2025) Bermúdez de Castro López-Varela, Alfredo; Gómez Pedreira, María Dolores; Martínez Suárez, Iván; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
The main objective of this paper is to compare the lumped parameter models commonly used in the calculation of Wireless Power Transfer (WPT) and the distributed model derived from Maxwell’s equations. First, the WPT between two coils in the harmonic regime is analyzed. A lumped parameter model for WPT between two coils is introduced, and theoretical calculations for power transfer efficiency (PTE) are obtained. Then, the main advantages and disadvantages of this model with respect to the Maxwell’s equations model are discussed and a procedure for parameter calculation is presented. A comparison of the two models is made through two numerical tests: the first representing the charging of a mobile phone and the second involving a wireless charging process of an electric vehicle (EV). Once the PTE is obtained for many frequency values, the calculations are compared, resulting in relevant errors committed by the lumped parameter model under certain conditions. The accuracy of the lumped model is particularly low when distributed eddy currents occur in the WPT problem because the model does not account for this phenomenon, as Maxwell’s equations do. Therefore, the Maxwell’s equations model should be used in such cases, despite its higher computational cost.
Multiphysics simulation of slag melting in an induction furnace for sustainable silicon production
(Elsevier, 2025-09) Bermúdez de Castro López-Varela, Alfredo; Crego Martínez, Óscar; Ferrín González, José Luis; García Correa, Branca; Gómez Pedreira, María Dolores; Martínez Suárez, Iván; Pérez Pérez, Luis Javier; Salgado Rodríguez, María del Pilar; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
This work presents a multiphysics mathematical modelling and numerical simulation of the slag melting process in an induction furnace, with a focus on the production of sustainable silicon through the EU SisAl Pilot project. The mathematical model incorporates electromagnetic, thermal and hydrodynamic phenomena in a coupled axisymmetric framework to simulate the melting of a CaO-SiO2 slag, a key component in the aluminothermic reduction process for silicon production. The model addresses the challenge of heating the poorly electrically conductive slag using a graphite crucible and it also accounts for buoyancy-driven convection in the molten slag. The numerical simulations are validated against experimental data from pilot scale trials at Elkem’s plant in Norway. In addition, sensitivity analyses are carried out considering both the progressive filling of the furnace and the inclusion of surface-to-surface radiation models.
A Finite Element Method for a Nonlinear Magnetostatic Problem in Terms of Scalar Potentials
(Wiley, 2025) Muñoz Sola, Rafael; Salgado Rodríguez, María del Pilar; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
The aim of this paper is to perform the analysis of a numerical method based on scalar potentials for solving a nonlinear magnetostatic problem in a three-dimensional bounded domain containing prescribed currents and magnetic materials. The method discretizes a well-known formulation of this problem based on two scalar potentials: the total potential, defined in magnetic materials, and the reduced potential, defined in dielectric media and in non-magnetic conductors carrying currents. The topology of the magnetic materials is not assumed to be trivial, which leads to a multivalued potential. The resulting nonlinear variational problem is proved to be well posed and is discretized by means of standard piecewise linear finite elements. A convergence result without regularity assumptions on the solutions is proved in both the linear and nonlinear cases. Moreover, optimal error estimates are proved for smooth functions. Numerical results for an analytical test are reported to assess the performance of the method in the case of the continuous solution being smooth.
IHP: a dynamic heterogeneous parallel scheme for iterative or time‑step methods—image denoising as case study
(Springer, 2021-01) Laso, Ruben; Cabaleiro Domínguez, José Carlos; Fernández Rivera, Francisco; Muñiz Castiñeira, María del Carmen; Álvarez Dios, José Antonio; Universidade de Santiago de Compostela. Centro de Investigación en Tecnoloxías Intelixentes da USC (CiTIUS); Universidade de Santiago de Compostela. Departamento de Matemática Aplicada; Universidade de Santiago de Compostela. Departamento de Electrónica e Computación
Iterative and time-step methods are spread far and wide in several mathematics and physics domains. At the same time, modern computers include multicore CPUs along with GPUs, so it is important to use all their computing capabilities for their efficient use. Aiming to improve performance of this kind of numerical methods, we introduce in this work a new heterogeneous parallelism CPU + GPU scheme which we call IHP. This new scheme has the advantage of being self-balanced and able to dynamically distribute the workload between CPU and GPU according to their performance on the fly. Also, it can be used with several contending technologies, like CUDA and OpenCL for GPUs or OpenMP and Intel TBB for CPUs. As a case in point, we analyse an image denoising problem based on time-step diffusion methods for brightness and chromaticity. Results show execution significant improvements in execution time using this scheme, with a minimal overhead.
Numerical modelling of a transient conductive-radiative thermal problem arising in silicon purification
(Elsevier, 2006-06-01) Bermúdez de Castro López-Varela, Alfredo; Leira, Rocío; Muñiz Castiñeira, María del Carmen; Pena Brage, Francisco José; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
The objective of this work is to present and numerically solve a mathematical model for the thermal behavior of a casting ladle devoted to purification of silicon. A nonlinear and non-local boundary condition is considered for radiative heat transfer in an inner closed cavity of the domain. We also propose a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented.
Mathematical and numerical study of transient wave scattering by obstacles with a new class of arlequin coupling
(Society for Industrial and Applied Mathematics, 2019) Albella Martínez, Jorge; Ben Dhia, H.; Imperiale, S.; Rodríguez García, Jerónimo; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
In this work, we extend the Arlequin method, a multiscale and multimodel framework based on overlapping domains and energy partitions for reliable modeling and flexible simulation of transient problems of wave scattering by obstacles. The main contribution is the derivation and analysis of new variants of the coupling operators. The constructed finite element and finite difference discretizations allow for solving wave propagation problems while using nonconforming and overlapping meshes for the background propagating medium and a local patch surrounding the obstacle, respectively. This provides a method with great flexibility and a low computational cost. The method is proved to be stable in terms of both space discrtization-an inf-sup condition is established-and time discretization-conservation of discrte energy is proved. 1 dimensional and 2 dimensional numerical results confirm the good perfomance of the overall discretization scheme.
Semi-implicit Hybrid Finite Volume/Finite Element Method for the GPR Model of Continuum Mechanics
(Springer, 2025-01-06) Busto Ulloa, Saray; Río-Martín, Laura; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
We present a new hybrid semi-implicit finite volume / finite element numerical scheme for the solution of incompressible and weakly compressible media. From the continuum mechanics model proposed by Godunov, Peshkov and Romenski (GPR), we derive the incompressible GPR formulation as well as a weakly compressible GPR system. As for the original GPR model, the new formulations are able to describe different media, from elastoplastic solids to viscous fluids, depending on the values set for the model’s relaxation parameters. Then, we propose a new numerical method for the solution of both models based on the splitting of the original systems into three subsystems: one containing the convective part and non-conservative products, a second subsystem for the source terms of the distortion tensor and thermal impulse equations and, finally, a pressure subsystem. In the first stage of the algorithm, the transport subsystem is solved by employing an explicit finite volume method, while the source terms are solved implicitly. Next, the pressure subsystem is implicitly discretised using continuous finite elements. This methodology employs unstructured grids, with the pressure defined in the primal grid and the rest of the variables computed in the dual grid. To evaluate the performance of the proposed scheme, a numerical convergence analysis is carried out, which confirms the second order of accuracy in space. A wide range of benchmarks is reproduced for the incompressible and weakly compressible cases, considering both solid and fluid media. These results demonstrate the good behaviour and robustness of the proposed scheme in a variety of scenarios and conditions.
An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems
(Society for Industrial and Applied Mathematics, 2024) Buffa, Annalisa; Chanon, Ondine Chanon; Grappein, Denise; Vázquez Hernández, Rafael; Vohralik, Martin; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
An a posteriori error estimator based on an equilibrated flux reconstruction is proposed for defeaturing problems in the context of finite element discretizations. Defeaturing consists in the simplification of a geometry by removing features that are considered not relevant for the approximation of the solution of a given PDE. In this work, the focus is on a Poisson equation with Neumann boundary conditions on the feature boundary. The estimator accounts both for the so-called defeaturing error and for the numerical error committed by approximating the solution on the defeatured domain. Unlike other estimators that were previously proposed for defeaturing problems, the use of the equilibrated flux reconstruction allows us to obtain a sharp bound for the numerical component of the error. Furthermore, it does not require the evaluation of the normal trace of the numerical flux on the feature boundary: this makes the estimator well suited for finite element discretizations, in which the normal trace of the numerical flux is typically discontinuous across elements. The reliability of the estimator is proven and verified on several numerical examples. Its capability to identify the most relevant features is also shown, in anticipation of a future application to an adaptive strategy.
A local regularity result for Neumann parabolic problems with nonsmooth data
(Elsevier, 2017) Martínez, A.; Muñoz Sola, Rafael; Vázquez Méndez, Miguel Ernesto; Álvarez-Vázquez, L. J.; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
In this work we analyze the relations between two different concepts of solution of the Neumann problem for a second order parabolic equation: the usual notions of weak solution and those of transposition solution, which allow well-posedness of problems with measure data. We give a regularity result for the transposition solution and we prove that, under smoothness assumptions for the principal part of the operator, the local regularity of the transposition solution is the same as that of the usual weak solution. As an interesting particular case, we present a rigorous proof of local continuity of the solution for a convection–diffusion problem with pointwise source term.
Mathematical analysis of a parabolic-elliptic problem with moving parabolic subdomain through a Lagrangian approach
(Elsevier, 2019) Muñoz Sola, Rafael; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
The aim of this paper is to study the regularity of the solution of some linear parabolic-elliptic problems in which parabolicity region depends on time. More specifically, this region is the position occupied by a body undergoing a motion (a deformation smoothly evolving in time). The main tool we introduce is a suitable extension of the motion to the entire spatial domain of the PDE. This enables us to reduce the original problem to a parabolic-elliptic problem with variable coefficients and with a parabolicity region independent of time. This problem can be seen as a Lagrangian formulation of our original problem. Next, we obtain regularity results for a class of parabolic-elliptic problems with variable coefficients and fixed parabolicity region. We apply these results to the Lagrangian formulation and, finally, we obtain a regularity result for our original problem.
An all Mach number semi-implicit hybrid Finite Volume/Virtual Element method for compressible viscous flows on Voronoi meshes
(Elsevier, 2025) Boscheri, Walter; Busto Ulloa, Saray; Dumbser, Michael; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
We present a novel high order semi-implicit hybrid finite volume/virtual element numerical scheme for the solution of compressible flows on Voronoi tessellations. The method relies on the operator splitting of the compressible Navier–Stokes equations into three sub-systems: a convective sub-system solved explicitly using a finite volume (FV) scheme, and the viscous and pressure sub-systems which are discretized implicitly with the aid of a virtual element method (VEM). Consequently, the time step restriction of the overall algorithm depends only on the mean flow velocity and not on the fast pressure waves nor on the viscous eigenvalues. As such, the proposed methodology is well suited for the solution of low Mach number flows at all Reynolds numbers. Moreover, the scheme is proven to be globally energy conserving so that shock capturing properties are retrieved in high Mach number flows while being only linearly implicit in time. To reach high order of accuracy in time and space, an IMEX Runge–Kutta time stepping strategy is employed together with high order spatial reconstructions in terms of CWENO polynomials and virtual element space basis functions. The chosen discretization techniques allow the use of general polygonal grids, a useful tool when dealing with complex domain configurations. The new scheme is carefully validated in both the incompressible limit and the high Mach number regime through a large set of classical benchmarks for fluid dynamics, assessing robustness and accuracy.
Adaptive methods with C1 splines for multi-patch surfaces and shells
(Elsevier, 2024) Bracco, Cesare; Farahat, Andrea; Giannelli, Carlotta; Kapl, Mario; Vázquez Hernández, Rafael; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
We introduce an adaptive isogeometric method for multi-patch surfaces and Kirchhoff–Love shell structures with hierarchical splines characterized by continuity across patches. We extend the construction of smooth hierarchical splines from the multi-patch planar setting to analysis suitable surfaces. The adaptive scheme to solve fourth order partial differential equations is presented in a general framework before showing its application for the numerical solution of the bilaplacian and the Kirchhoff–Love model problems. A selection of numerical examples illustrates the performance of hierarchical adaptivity on different multi-patch surface configurations.
Tree–cotree-based tearing and interconnecting for 3D magnetostatics: A dual–primal approach
(Elsevier, 2025) Mally, Mario; Kapidani, Bernard; Merkel, Melina; Schöps, Sebastian; Vázquez Hernández, Rafael; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada
The simulation of electromagnetic devices with complex geometries and large-scale discrete systems benefits from advanced computational methods like IsoGeometric Analysis and Domain Decomposition. In this paper, we employ both concepts in an Isogeometric Tearing and Interconnecting method to enable the use of parallel computations for magnetostatic problems. We address the underlying non-uniqueness by using a graph-theoretic approach, the tree–cotree decomposition. The classical tree–cotree gauging is adapted to be feasible for parallelization, which requires that all local subsystems are uniquely solvable. Our contribution consists of an explicit algorithm for constructing compatible trees and combining it with a dual–primal approach to enable parallelization. The correctness of the proposed approach is proved and verified by numerical experiments, showing its accuracy, scalability and optimal convergence.
A mathematical tool for road design projects
(Taylor & Francis, 2024) Vázquez Méndez, Miguel Ernesto; Blanco-Valcarcel, Pedro; Casal Urcera, Gerardo; Castro Ponte, Alberte; Santamarina Ríos, Duarte; Universidade de Santiago de Compostela. Departamento de Matemática Aplicada; Universidade de Santiago de Compostela. Departamento de Enxeñaría Agroforestal
This article introduces a mathematical tool for the semi-automatic design of multiple layout alternatives for a road, and the selection, between all of them, of the most suitable option in each particular case study. This tool is based on a mathematical programming model that includes the required technical constraints to ensure safe and comfortable driving (fulfilling the current Spanish legislation) and automates the process of selecting the connecting points to the former road. This optimization module is completed by two other modules: a clustering one, that groups the layouts obtained in corridors, and a decision-making aid one, which assists the engineer in the selection of a layout alternative in each corridor and, subsequently, in the choice of the optimal overall layout. The numerical results show the usefulness of this tool in a real-life case study that consists in designing a bypass on the Spanish road N-640 that circumvents the urban area of Meira (Lugo).

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