Research topics and possible TFG/TFM directions for undergraduate and master students.
Computational electromagnetics
This line focuses on numerical methods for solving Maxwell’s equations in complex scenarios. The main technical core is the finite element method (FEM), especially high-order curl-conforming formulations, Nedelec basis functions, domain decomposition methods, verification, and large-scale simulation. Strongly related with Luis E. García Castillo and Sergio Llorente Romano.
Implementing and testing FEM solvers in MATLAB, Julia, Python, Fortran, or JAX.
Developing automatic verification tests for electromagnetic simulation codes.
Working with mesh generation, mesh orientation, and interfaces with tools such as Gmsh, Cubit, GiD, or ParaView.
Studying domain decomposition methods for large electromagnetic problems.
Exploring high-order basis functions for tetrahedra, prisms, and hexahedra.
This line connects with previous work on high-order FEM, Nedelec elements, domain decomposition, code verification through manufactured solutions, and industrial electromagnetic simulation.
Selected references and conference contributions:
selected papers
2025
A priori verification method for curl-conforming basis functions in simplices
Adrian Amor-Martin, and Luis E. Garcia-Castillo
Mathematical Methods in the Applied Sciences, 2025
The construction of (hierarchical) curl-conforming basis functions has been a hot topic in the last decades in the finite element community. Especially, functions applied to simplices have been quite popular after the work by Nédélec in 1980. Many mixed-order and full-order families have been provided in the last years, but sometimes, it is difficult to assess if they belong to the original space proposed by Nédélec (especially when orthogonalization procedures are applied). Here, a tool to determine if a family of basis functions belongs to the Nédélec space is provided. Since affine coordinates are the most frequent choice for simplices, particularities about its use with this kind of coordinates are detailed. A detailed survey of existing families is provided, and the practical application of the tool to a representative set of these families is discussed. The tool is also available for the community in a public repository.
2024
A Rigorous Code Verification Process of the Domain Decomposition Method in a Finite Element Method For Electromagnetics
Adrian Amor-Martin, Luis E. Garcia-Castillo, Laszlo L. Toth, and 2 more authors
IEEE Transactions on Antennas and Propagation, 2024
The finite element method (FEM) has benefited recently from the introduction of the domain decomposition method (DDM), especially to tackle large-scale problems. Many versions of DDM may be found in the literature, and almost none of them have used a rigorous approach to show the accuracy of the formulation. Here, we suggest the use of the method of manufactured solutions (MMS) to introduce DDM into a FEM code. This approach also allows us to debug the demanding coding process of the method. First, we introduce the mandatory operators used for DDM to construct second-order absorbing boundary conditions. Then, we use these results to show the correct implementation of DDM with basis functions up to fourth order for tetrahedra, triangular prisms, and hexahedra, obtaining the convergence rates predicted by the classic FEM theory. Finally, we illustrate how to use MMS to test different formulations, assessing the effect of using different spaces and orders for the arising ancillary variables.
Hierarchical Universal Matrices for Curvilinear Tetrahedral H(Curl) Finite Elements with Inhomogeneous Material Properties
Laszlo L. Toth, Adrian Amor-Martin, and Romanus Dyczij-Edlinger
IEEE Transactions on Antennas and Propagation, 2024
A general method for calculating the mass and stiffness matrices of H(curl)-conforming finite elements is pro- posed. It applies to curvilinear geometries geometry and/or inhomogeneous materials, preserves the nullspace of the curl operator, and is computationally efficient. In the finite element integrals, the terms incorporating the effects of geometry and materials are expanded in a series of multivariate polynomials. As a result, the element matrices can always be integrated analytically by means of predefined universal matrices. The mapping from the reference element to the physical configuration is done performed by suitable forms of the Piola transformation the Piola transformations so that the resulting finite element bases are guaranteed to preserve the nullspace of the curl operator. A mathematical proof, including the curvilinear case, is included. The suggested approach features a representation limit: When the metric expansion is continued beyond a critical order, which is determined by the order of the finite element basis, the truncation error becomes zero. Hence, the number of universal matrices is bounded, and the sole source of error is the numerical calculation of the expansion coefficients for the metric terms. The method is validated by numerical examples, and its computational efficiency is demonstrated by a comparison to competing approaches.
2023
Second-Order Nédélec Curl-Conforming Hexahedral Element for Computational Electromagnetics
Adrian Amor-Martin, and Luis E. Garcia-Castillo
IEEE Transactions on Antennas and Propagation, 2023
We follow a systematic approach to obtain mixed-order curl-conforming basis functions for the hexahedron that are compatible with basis functions for tetrahedra and triangular prisms previously published. The approach is mathematically sound since we obtain the functions as the dual basis with respect to properly discretized Nédélec degrees of freedom. Simple to use and well-conditioned bases without the need for added orthogonalization procedures are obtained. We provide simple closed-form expressions for second-order basis functions in a reference hexahedron in terms of integer coefficients and monomials. The expressions are ready to use as long as the appropriate geometric mappings are made. We apply the Method of Manufactured Solutions to a finite element double curl vector wave formulation for verification purposes; specifically, we make a study of the non-symmetrical structure of the corresponding tensor product finite element space. We also solve generalized eigenvalue problems for well-known cavities. We provide the open-source code for generating the coefficients, evaluating the basis functions, and computing the finite element matrices involved in some of the numerical solutions shown in the paper.
2021
Study of Accuracy of a Non-Conformal Finite Element Domain Decomposition Method
Adrian Amor-Martin, Luis E. Garcia-Castillo, and Jin-Fa Lee
Domain Decomposition Methods (DDM) have been widely used in the Computational Electromagnetics (CEM) community in the last years to tackle large-scale problems with Finite Element Methods (FEM). Non-conformal DDM is more flexible (e.g., independently created meshes for different parts of the problem under analysis are supported) but may introduce an approximation error. In this communication, a thorough study of the accuracy of the solutions when using non-conformal DDM is presented. Three experiments are realized showing the verification of the implementation and that the accuracy is acceptable with different numbers of discontinuities in the propagation direction and various aspect ratios of the mesh on the interface. The numerical results use three different shapes (tetrahedra, prisms, and hexahedra) and up to order three in the basis functions that approximate the field. These studies are relevant for the introduction of non-conformal DDM with real problems or scalable (in the parallel sense) implementation of adaptive mesh techniques.
2016
Second-Order Nédélec Curl-Conforming Prismatic Element for Computational Electromagnetics
Adrian Amor-Martin, Luis E. Garcia-Castillo, and Daniel Garcia-Donoro
IEEE Transactions on Antennas and Propagation, 2016
A systematic approach to obtaining mixed-order curl-conforming basis functions for a triangular prism is presented; focus is made on the second-order case. Space of functions for the prism is given. Basis functions are obtained as the dual basis with respect to suitably discretized Nédélec degrees of freedom functionals acting on elements of the space. Thus, the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed. Different strategies for the finite element assembly of the basis are discussed. Numerical results showing the verification procedure of the correctness of the implemented basis functions are given. Numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown. Comparison with other representative sets of basis functions for prisms is included.
conference contributions
2025
Anisotropic Nédélec Curl-Conforming Prismatic Element
Adrian Amor-Martin, Sergio Llorente-Romano, and Luis E. Garcia-Castillo
In 25th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, 2025
This line explores the intersection between electromagnetic simulation and AI. The goal is to build models that can infer material properties, detect anomalies, or accelerate inverse problems by combining physics-based solvers with machine learning.
Developing differentiable electromagnetic solvers with JAX or other scientific computing tools.
Training AI models using synthetic data generated by FEM or microwave simulations.
Detecting biological tissue anomalies or material properties from RF/microwave responses.
Comparing automatic differentiation tools for scientific computing.
Building pipelines that connect simulation, data labeling, training, and validation.
Selected references:
conference contributions
2025
Differentiable Solvers for Electromagnetics. At the Intersection of Scientific Computing and Machine Learning
Eduardo Gómez-González, Luis E. Garcia-Castillo, Sergio Llorente-Romano, and 1 more author
In XVI Encuentro Ibérico de Electromagnetismo Computacional, 2025
Geo-electromagnetics
This line applies computational electromagnetics to subsurface exploration, geophysical modeling, and electromagnetic inverse problems at low frequencies. It includes collaboration with researchers working on large-scale scientific computing and geophysical simulations.
Preparing datasets from electromagnetic simulations for AI-based classification.
We present numerical experiments for geophysics electromagnetic (EM) modeling based upon high-order edge elements and supervised h+p refinement approaches on massively parallel computers. Our high-order h+p refinement strategy is based on and extends the PETGEM code. We focus on the performance study in terms of accuracy, convergence rate, and computational effort to solve real-life 3D setups based on synthetic and experimental data for energy reservoir characterization. These test cases show variable resolution discretization needs and realistic physical parameters. In general, our numerical results are consistent theoretically. The use of h-adapted meshes was efficient to achieve a certain accuracy level in the synthetic EM responses. Regarding global p-refinement, p=2 exhibits the best accuracy/performance trade-off. Selective p-refinement might offer a better compromise between accuracy and computational cost. However, for p-refinement at different entities, the best refinement scheme consists of using p=3 at the volume level with p=1 at faces and edges. Thus, p-refinement can be competitive if applied hierarchically. Nevertheless, we acknowledge that the performance of our supervised h+p refinement strategy depends on the input model (e.g., conductivity, frequency, domain decomposition strategy, among others). Whatever the chosen configuration, our numerical results provide an in-depth understanding of EM modeling’s pros and cons when supervised h+p refinement schemes are applied.
3D Magnetotelluric Modeling Using High-Order Tetrahedral Nédélec Elements on Massively Parallel Computing Platforms
Octavio Castillo-Reyes, David Modesto, Pilar Queralt, and 5 more authors
We present a routine for 3D magnetotelluric (MT) modeling based upon high-order edge finite element method (HEFEM), tailored and unstructured tetrahedral meshes, and high-performance computing (HPC). This implementation extends the PETGEM modeller capabilities, initially developed for active-source electromagnetic methods in frequency-domain. We assess the accuracy, robustness, and performance of the code using a set of reference models developed by the MT community in well-known reported workshops. The scale and geological properties of these 3D MT setups are challenging, making them ideal for addressing a rigorous validation. Our numerical assessment proves that this new algorithm can produce the expected solutions for arbitrarily 3D MT models. Also, our extensive experimental results reveal four main insights: (1) high-order discretizations in conjunction with tailored meshes can offer excellent accuracy; (2) a rigorous mesh design based on the skin-depth principle can be beneficial for the solution of the 3D MT problem in terms of numerical accuracy and run-time; (3) high-order polynomial basis functions achieve better speed-up and parallel efficiency ratios than low-order polynomial basis functions on cutting-edge HPC platforms; (4) a triple helix approach based on HEFEM, tailored meshes, and HPC can be extremely competitive for the solution of realistic and complex 3D MT models and geophysical electromagnetics in general.
Heterogeneous and embedded computing
This line studies how scientific and AI workloads behave on GPUs, embedded platforms, and heterogeneous architectures. The focus is on performance, energy efficiency, reliability, and deployment constraints. Strongly related to José Antonio Belloch Rodríguez.
Benchmarking AI models on embedded GPU platforms.
Accelerating simulation kernels or signal-processing workloads.
Comparing CPU, GPU, and embedded implementations.
Evaluating quantization, TensorRT, and other deployment optimizations.
Studying reliability issues in edge AI systems.
This line connects with work on GPU acceleration, many-core architectures, edge AI, embedded deep learning, and reliability of neural networks in demanding environments.
Selected references:
selected papers
2026
Real-Time Object Tracking with on-Device Deep Learning for Adaptive Beamforming in Dynamic Acoustic Environments
Jorge Ortigoso-Narro, Jose A. Belloch, Adrian Amor-Martin, and 2 more authors
Advances in object tracking and acoustic beamforming are driving new capabilities in surveillance, human-computer interaction, and robotics. This work presents an embedded system that integrates deep learning–based tracking with beamforming to achieve precise sound source localization and directional audio capture in dynamic environments. The approach combines single-camera depth estimation and stereo vision to enable accurate 3D localization of moving objects. A planar concentric circular microphone array constructed with MEMS microphones provides a compact, energy-efficient platform supporting 2D beam steering across azimuth and elevation. Real-time tracking outputs continuously adapt the array’s focus, synchronizing the acoustic response with the target’s position. By uniting learned spatial awareness with dynamic steering, the system maintains robust performance in the presence of multiple or moving sources. Experimental evaluation demonstrates significant gains in signal-to-interference ratio, making the design well-suited for teleconferencing, smart home devices, and assistive technologies.
2025
Enhanced U-Net Architectures for Accurate Room Impulse Response Generation via Differential-Phase Learning
Ignacio Martin-Salinas, Gema Piñero, Jose A. Belloch, and 1 more author
EURASIP Journal on Audio, Speech, and Music Processing, Feb 2025
Generating accurate room impulse responses (RIRs) remains challenging, particularly regarding phase estimation. Building upon previous work utilizing encoder-decoder deep learning architectures, this paper investigates advanced techniques to improve phase prediction accuracy. We propose and evaluate several enhanced U-Net models, including variants with a variational autoencoder (VAE) bottleneck and differing input conditioning methods for spatial and room parameters (embedding layers vs. normalized dense layers). A key focus is the comparison between predicting direct phase and differential phase. Furthermore, we analyze the impact of using mean absolute error (MAE) versus mean squared error (MSE) for the magnitude component of the loss function. The study also explores the efficacy of applying the Griffin-Lim algorithm as a post-processing step to refine the phase estimated by the networks. Performance is evaluated on a real RIR dataset, comparing the different model architectures, information vector encoding strategies, phase targets (direct vs. differential), loss functions, and the contribution of phase recovery algorithms to overall RIR fidelity. Results provide insights into effective strategies for enhancing phase generation in data-driven RIR synthesis.
Reliability of Vision Transformers and CNNs on Edge AI Systems under Neutron Radiation
Jose M. Badia, Ignacio Martin-Salinas, German Leon, and 7 more authors
The reliability of neural networks on Edge AI platforms is crucial for safety-critical applications, especially in environments exposed to radiation. This paper presents an experimental evaluation of the reliability of Vision Transformers (ViTs) and Convolutional Neural Networks (CNNs) deployed on low-power System-on-Chip (SoC) devices, specifically the Jetson Orin Nano. The study explores the impact of neutron radiation on these models, focusing on the effect of optimisation techniques such as TensorRT, data precision reduction, and weight quantisation. Results indicate that while optimisation can improve inference performance, it may also increase the error rates in outputs. Additionally, including main memory in the radiation tests resulted in severe persistent errors, highlighting the need for thorough reliability assessments in real-world scenarios. A comparison between ViTs and CNNs revealed that ViTs, despite their complexity, show comparable or better reliability on modern SoCs, which is critical for deploying AI in safety-critical environments like aerospace and autonomous driving.
2020
GPU Acceleration of a Non-Standard Finite Element Mesh Truncation Technique for Electromagnetics
José M. Badía, Adrian Amor-Martin, Jose A. Belloch, and 1 more author
The emergence of General Purpose Graphics Processing Units (GPGPUs) provides new opportunities to accelerate applications involving a large number of regular computations. However, properly leveraging the computational resources of graphical processors is a very challenging task. In this paper, we use this kind of device to parallelize FE-IIEE (Finite Element-Iterative Integral Equation Evaluation), a non-standard finite element mesh truncation technique introduced by two of the authors. This application is computationally very demanding due to the amount, size and complexity of the data involved in the procedure. Besides, an efficient implementation becomes even more difficult if the parallelization has to maintain the complex workflow of the original code. The proposed implementation using CUDA applies different optimization techniques to improve performance. These include leveraging the fastest memories of the GPU and increasing the granularity of the computations to reduce the impact of memory access. We have applied our parallel algorithm to two real radiation and scattering problems demonstrating speedups higher than 140 on a state-of-the-art GPU.
Antennas, microwave sensors, and RF prototypes
This line is closer to hardware, measurement, and RF design. It includes antennas, microwave sensors, active resonant circuits, biological and material characterization, and experimental validation. Strongly related with Daniel Segovia Vargas.
Designing and measuring antennas or antenna arrays.
Developing microwave sensors for material characterization.
Studying active sensors based on oscillators or amplifiers.
Simulating biological phantoms and tissue-detection scenarios.
Co-simulating active and passive RF circuits with tools such as ADS or CST.
Fabricating and testing microwave circuits for research or teaching laboratories.
This line is a strong option for students who want to combine simulation with laboratory work.
Selected references and conference contributions:
selected papers
2026
Stability Considerations for the Design of Amplifier-Based Active Sensors
Sandra Santiago-Mesas, Adrian Amor-Martin, Vicente González-Posadas, and 1 more author
This paper presents a systematic design methodology for amplifier-based active sensors, with stability as a central consideration. The proposed sensor consists of a passive resonant structure and an active element connected in series to form a single-port amplifier operating in reflection. A lumped-element equivalent circuit was first employed to determine an initial value for the impedance of the active element, which was used as starting point for the design using realistic models of a Complementary Split-Ring Resonator (CSRR), for the passive resonant structure, and a Clapp oscillator, for the active element. The Clapp oscillator was optimized until until a threefold improvement in the quality factor compared to the passive case without compromising the stability of the system. The sensing performance of the proposed active sensor was validated both in simulation and measurements on lossy samples, showing improved performance without compromising its sensitivity. System stability was confirmed by examining the NDF (in simulation) and the spectral response (in simulation and measurement). Despite promising results were obtained, further optimization of the active element is required to increase the stability margins and the quality factor. Moreover, the integration of both the passive resonant structure and the active element subsystems into a single compact circuit, as well as the analysis of nonlinear effects, should be addressed in future works.
conference contributions
2024
High-Stability Oscillator-Based Sensor for Low-Cost Biological Phantom Validation
Sandra Santiago-Mesas, Elizabeth Fernandez-Aranzamendi, Adrian Amor-Martin, and 2 more authors
In 2024 IEEE MTT-S International Microwave Biomedical Conference (IMBioC), 2024
Active Sensor Design Based on Large-Signal Stability Analysis with Pole-Zero Identification
Sandra Santiago-Mesas, Elizabeth Fernández-Aranzamendi, Adrián Amor-Martín, and 2 more authors
In 2024 54th European Microwave Conference (EuMC), 2024
2023
A High-Stability and High-Sensitivity Active Sensor for Non-Invasive Breast Cancer Detection
Sandra Santiago-Mesas, Elizabeth Fernandez-Aranzamendi, Daniel Segovia-Vargas, and 2 more authors
In 53rd European Microwave Conference, 2023
ISAC, communications, audio, and XR
This line covers application-driven projects around sensing and communications, including integrated sensing and communications, adaptive beamforming, embedded AI, XR/360-degree media, and quality of experience (strongly related with Marta Orduna, from Nokia)
Typical student projects include:
Exploring 5G/6G sensing and communication scenarios.
Working on adaptive beamforming with embedded AI.
Developing prototypes for antenna arrays or acoustic arrays.
Applying deep learning to XR, 360-degree video, or audio selection.
Studying measurement-driven quality of experience in immersive systems.
Available positions for students
You do not need to arrive as an expert. A good project usually starts with curiosity, consistency, and a willingness to learn the tools. Depending on the topic, useful skills include programming, linear algebra, electromagnetics, RF design, machine learning, scientific computing, or laboratory measurements.
Some projects are more mathematical, some are more software-oriented, and some involve hardware and experiments. The scope can be adapted to TFG or TFM level.
For concrete available topics, see the TFG/TFM page. For the research background behind these lines, see the publications page.
