Construction of higher-order curl-conforming finite elements and its assembly

Different choices are available when constructing vector finite element bases in real coordinates. In this communication, two different designs of higher-order curl-conforming basis functions are introduced and explained, showing the particularities of its assembly. Tetrahedra and hexahedra are used as element shapes to assess the effect of triangular and quadrilateral faces on the two considered constructions of basis functions. A comparison of their robustness in terms of the condition number of the finite element matrices for a number of distortions is includedMinisterio de Ciencia y Tecnología, Grant/Award Numbers: TEC2013- 47753-C3, TEC2016-80386-P; Ministerio de Educación, Cultura y Deporte, Grant/Award Number: FPU14/0374

Adaptive Semi-Structured Mesh Refinement Techniques for the Finite Element Method

The adaptive mesh techniques applied to the Finite Element Method have continuously been an active research line. However, these techniques are usually applied to tetrahedra. Here, we use the triangular prismatic element as the discretization shape for a Finite Element Method code with adaptivity. The adaptive process consists of three steps: error estimation, marking, and refinement. We adapt techniques already applied for other shapes to the triangular prisms, showing the differences here in detail. We use five different marking strategies, comparing the results obtained with different parameters. We adapt these strategies to a conformation process necessary to avoid hanging nodes in the resulting mesh. We have also applied two special rules to ensure the quality of the refined mesh. We show the effect of these rules with the Method of Manufactured Solutions and numerical results to validate the implementation introduced.This work has been financially supported by TEC2016-80386-

Parallel 3-D marine controlled-source electromagnetic modelling using high-order tetrahedral Nédélec elements

We present a parallel and high-order Nédélec finite element solution for the marine controlled-source electromagnetic (CSEM) forward problem in 3-D media with isotropic conductivity. Our parallel Python code is implemented on unstructured tetrahedral meshes, which support multiple-scale structures and bathymetry for general marine 3-D CSEM modelling applications. Based on a primary/secondary field approach, we solve the diffusive form of Maxwell’s equations in the low-frequency domain. We investigate the accuracy and performance advantages of our new high-order algorithm against a low-order implementation proposed in our previous work. The numerical precision of our high-order method has been successfully verified by comparisons against previously published results that are relevant in terms of scale and geological properties. A convergence study confirms that high-order polynomials offer a better trade-off between accuracy and computation time. However, the optimum choice of the polynomial order depends on both the input model and the required accuracy as revealed by our tests. Also, we extend our adaptive-meshing strategy to high-order tetrahedral elements. Using adapted meshes to both physical parameters and high-order schemes, we are able to achieve a significant reduction in computational cost without sacrificing accuracy in the modelling. Furthermore, we demonstrate the excellent performance and quasi-linear scaling of our implementation in a state-of-the-art high-performance computing architecture.This project has received funding from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie grant agreement No. 777778. Furthermore, the research leading to these results has received funding from the European Union's Horizon 2020 programme under the ChEESE Project (https://cheese-coe.eu/ ), grant agreement No. 823844. In addition, the authors would also like to thank the support of the Ministerio de Educación y Ciencia (Spain) under Projects TEC2016-80386-P and TIN2016-80957-P. The authors would like to thank the Editors-in-Chief and to both reviewers, Dr. Martin Cuma and Dr. Raphael Rochlitz, for their valuable comments and suggestions which helped to improve the quality of the manuscript. This work benefited from the valuable suggestions, comments, and proofreading of Dr. Otilio Rojas (BSC). Last but not least, Octavio Castillo-Reyes thanks Natalia Gutierrez (BSC) for her support in CSEM modeling with BSIT.Peer ReviewedPostprint (author's final draft

An interface between an hp-adaptive finite element package and the pre- and post-processor GiD

An interface between GiD, the interactive graphical user interface used for numerical simulations, developed at the International Center for Numerical Methods in Engineering (CIMNE) of the Universidad Politécnica de Cataluña and the Geometrical Modeling Package (GMP) of the fully automatic hp-adaptive finite element (FE) software, developed at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin, is presented. GiD is used to construct a tessellation of the problem domain into FE-like regions (blocks in GMP terminology), and the interface obtains and transfers all the topological and geometrical information to GMP. Then, GMP automatically constructs a parameterization for each FE-like region of the GMP mesh, which later can be used to generate the actual FE-mesh and support geometry updates during mesh refinements

Método de Elementos Finitos hp con Adaptabilidad Automática Orientada a un Objetivo para Problemas Abiertos en 2D

In this paper, we describe a fully automatic goaloriented hp-adaptive Finite Element strategy, which is applied to open problems (radiation and scattering). The methodology produces exponential convergence rates in terms of an upper bound of an user-prescribed quantity of interest (in our case, the S-parameter, the far radiated field or far scattering field) against the problem size (number of degrees of freedom). We illustrate the efficiency of the method with 2D numerical simulations of open problems (radiation and scattering). Applications include the far scattering (radiated) field by an object (antenna) and the computation of mutual coupling of the antennas (S-parameters). Results show that self-adaptive goal-oriented hp obtains more accuracy in the quantity of interest than self-adaptive energynorm hp with the same number of degrees of freedom

Second-Order Nedelec Curl-Conforming Prismatic Element for Computational Electromagnetics

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 Nedelec 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.This work was supported by "DiDaCTIC: Desarrollo de un sistema de comunicaciones inalambrico en rango THz integrado de alta tasa de datos"; TEC2013-47753-C3, CAM S2013/ICE-3004 "DIFRAGEOS" projects and "Ayudas para contratos predoctorales de Formación del Profesorado Universitario FPU

A Nonstandard Schwarz Domain Decomposition Method for Finite-Element Mesh Truncation of Infinite Arrays

A nonstandard Schwarz domain decomposition method is proposed as finite-element mesh truncation for the analysis of infinite arrays. The proposed methodology provides an (asymptotic) numerically exact radiation condition regardless of the distance to the sources of the problem and without disturbing the original sparsity of the finite-element matrices. Furthermore, it works as a multi Floquet mode (propagating and evanescent) absorbing boundary condition. Numerical results illustrating main features of the proposed methodology are shown.This work was supported in part by the National Key Research and Development Program of China under Grant 2016YFE0121600, in part by the China Postdoctoral Science Foundation under Grant 2017M613068, in part by the National Key Research and Development Program of China under Grant 2017YFB0202102, and in part by the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund under Grant U1501501

Test-Driven Development of a Substructuring Technique for the Analysis of Electromagnetic Finite Periodic Structures

In this paper, we follow the Test-Driven Development (TDD) paradigm in the development of an in-house code to allow for the finite element analysis of finite periodic type electromagnetic structures (e.g., antenna arrays, metamaterials, and several relevant electromagnetic problems). We use unit and integration tests, system tests (using the Method of Manufactured Solutions—MMS), and application tests (smoke, performance, and validation tests) to increase the reliability of the code and to shorten its development cycle. We apply substructuring techniques based on the definition of a unit cell to benefit from the repeatability of the problem and speed up the computations. Specifically, we propose an approach to model the problem using only one type of Schur complement which has advantages concerning other substructuring techniques.This work has been financially supported by TEC2016-80386-P and PID2019-109984RB-C41

High-accuracy adaptive simulations of a Petri dish exposed to electromagnetic radiation

This paper analyses numerically the electric field distribution of a liquid contained in a Petri dish when exposed to electromagnetic waves excited in a rectangular waveguide. Solutions exhibit high-gradients due to the presence of the dielectric liquid contained in the dish. Furthermore, electromagnetic fields within the dielectric have a dramatically lower value than on the remaining part of the domain, which difficults its simulation. Additionally, various singularities of different intensity appear along the boundary of the Petri dish. To properly reproduce and numerically study those effects, we employ a highly-accurate hp-adaptive finite element method. Results of this study demonstrate that the electric field generated within the circular Petri dish is non-homogeneous, and thus, a better shape, size, or location of the dish is needed to achieve an equally distributed radiation enabling the uniform growth of cell cultives

Cálculo de Dosimetría Mediante Elementos Finitos con Adaptabilidad Automática hp en Tres Dimensiones

In this communication the effect of the electromagnetic radiation on in vitro cell cultures is analyzed using a self-adaptive hp-Finite Element Method (hp-FEM) in three dimensions. Computer dosimetry is a challenging problem as it involves complex geometries with high contrast electromagnetic materials. hp-FEM produces exponential convergence rates in terms of the energy-norm error of the solution against the problem size (number of degrees of freedom), even in the presence of singularities. Thus, accurate electromagnetic solutions of complex problems, as computer dosimetry for in vitro experiments, can be obtained. Furthermore, the use of self-adaptive techniques provides solutions, with a user pre-specified degree of accuracy from a initial very coarse mesh without any a priori knowledge of the problem solutio