Adrián Amor Martín

2025

1. 

   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

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   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

1. 

   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

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   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.

2023

1. 

   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

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   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.

2022

1. 

   Tailored Meshing for Parallel 3D Electromagnetic Modeling Using High-Order Edge Elements

   Octavio Castillo-Reyes, Adrian Amor-Martin, Arnaud Botella, and 2 more authors

   Journal of Computational Science, Aug 2022

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   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.