Self-adaptive hp finite element method with iterative mesh truncation technique accelerated with Adaptive Cross Approximation

ABSTRACT To alleviate the computational bottleneck of a powerful two-dimensional self-adaptive hp finite element method (FEM) for the analysis of open region problems, which uses an iterative computation of the Integral Equation over a fictitious boundary for truncating the FEM domain, we propose the use of Adaptive Cross Approximation (ACA) to effectively accelerate the computation of the Integral Equation. It will be shown that in this context ACA exhibits a robust behavior, yields good accuracy and compression levels up to 90%, and provides a good fair control of the approximants, which is a crucial advantage for hp adaptivity. Theoretical and empirical results of performance (computational complexity) comparing the accelerated and non-accelerated versions of the method are presented. Several canonical scenarios are addressed to resemble the behavior of ACA with h, p and hp adaptive strategies, and higher order methods in general

Goal-oriented self-adaptive hp-strategies for finite element analysis of electromagnetic scattering and radiation problems

In this paper, a fully automatic goal-oriented hp-adaptive finite element strategy for open region electromagnetic problems (radiation and scattering) is presented. The methodology leads to exponential rates of convergence in terms of an upper bound of an user-prescribed quantity of interest. Thus, the adaptivity may be guided to provide an optimal error, not globally for the field in the whole finite element domain, but for specific parameters of engineering interest. For instance, the error on the numerical computation of the S-parameters of an antenna array, the field radiated by an antenna, or the Radar Cross Section on given directions, can be minimized. The efficiency of the approach is illustrated with several numerical simulations with two dimensional problem domains. Results include the comparison with the previously developed energy-norm based hp-adaptivity

GPU Acceleration of a Non-Standard Finite Element Mesh Truncation Technique for Electromagnetics

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.This work was supported in part by the Spanish Government under Grant TEC2016-80386-P, Grant TIN2017-82972-R, and Grant ESP2015-68245-C4-1-P, and in part by the Valencian Regional Government under Grant PROMETEO/2019/109

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

Advanced techniques in scientific computing: application to electromagnetics

Mención Internacional en el título de doctorDurante los últimos años, los componentes de radiofrecuencia que forman parte de un sistema de comunicaciones necesitan simulaciones cada vez más exigentes desde el punto de vista de recursos computacionales. Para ello, se han desarrollado diferentes técnicas con el método de los elementos finitos (FEM) como la conocida como adaptatividad hp, que consiste en estimar el error en el problema electromagnético para generar mallas de elementos adecuadas al problema que obtienen una aproximación de forma más efectiva que las mallas estándar; o métodos de descomposición de dominios (DDM), basado en la división del problema original en problemas más pequeños que se pueden resolver en paralelo. El principal problema de las técnicas de adaptatividad es que ofrecen buenas prestaciones para problemas bidimensionales, mientras que en tres dimensiones el tiempo de generación de las mallas adaptadas es prohibitivo. Por otra parte, DDM se ha utilizado satisfactoriamente para la simulación de problemas eléctricamente muy grandes y de gran complejidad, convirtiéndose en uno de los temas más actuales en la comunidad de electromagnetismo computacional. El principal objetivo de este trabajo es estudiar la viabilidad de algoritmos escalables (en términos de paralelización) combinando DDM no conformes y adaptatividad automática en tres dimensiones. Esto permitir ía la ejecución de algoritmos de adaptatividad independiente en cada subdominio de DDM. En este trabajo se presenta y discute un prototipo que combina técnicas de adaptatividad y DDM, que aún no se han tratado en detalle en la comunidad científica. Para ello, se implementan tres bloques fundamentales: i) funciones de base para los elementos finitos que permitan órdenes variables dentro de la misma malla; ii) DDM no conforme y sin solapamiento; y iii) algoritmos de adaptatividad en tres dimensiones. Estos tres bloques se han implementado satisfactoriamente en un código FEM mediante un método sistemático basado en el método de las soluciones manufacturadas (MMS). Además, se ha llevado a cabo una paralelización a tres niveles: a nivel de algoritmo, con DDM; a nivel de proceso, con MPI (Message Passing Interface); y a nivel de hebra, con OpenMP; todo en un código modular que facilita el mantenimiento y la introducción de nuevas características. Con respecto al primer bloque fundamental, se ha desarrollado una familia de funciones base con un enfoque sistemático que permite la expansión correcta del espacio de funciones. Por otra parte, se han introducido funciones de base jerárquicas de otros autores (con los que el grupo al que pertenece el autor de la tesis ha colaborado estrechamente en los últimos años) para facilitar la introducción de diferentes órdenes de aproximación en el mismo mallado. En lo relativo a DDM, se ha realizado un estudio cuantitativo del error generado por las disconformidades en la interfaz entre subdominios, incluidas las discontinuidades generadas por un algoritmo de adaptatividad. Este estudio es fundamental para el correcto funcionamiento de la adaptatividad, y no ha sido evaluado con detalle en la comunidad científica. Además, se ha desarrollado un algoritmo de adaptatividad con prismas triangulares, haciendo especial énfasis en las peculiaridades debidas a la elección de este elemento. Finalmente, estos tres bloques básicos se han utilizado para desarrollar, y discutir, un prototipo que une las técnicas de adaptatividad y DDM.In the last years, more and more accurate and demanding simulations of radiofrequency components in a system of communications are requested by the community. To address this need, some techniques have been introduced in finite element methods (FEM), such as hp adaptivity (which estimates the error in the problem and generates tailored meshes to achieve more accuracy with less unknowns than in the case of uniformly refined meshes) or domain decomposition methods (DDM, consisting of dividing the whole problem into more manageable subdomains which can be solved in parallel). The performance of the adaptivity techniques is good up to two dimensions, whereas for three dimensions the generation time of the adapted meshes may be prohibitive. On the other hand, large scale simulations have been reported with DDM becoming a hot topic in the computational electromagnetics community. The main objective of this dissertation is to study the viability of scalable (in terms of parallel performance) algorithms combining nonconformal DDM and automatic adaptivity in three dimensions. Specifically, the adaptivity algorithms might be run in each subdomain independently. This combination has not been detailed in the literature and a proof of concept is discussed in this work. Thus, three building blocks must be introduced: i) basis functions for the finite elements which support non-uniform approximation orders p; ii) non-conformal and non-overlapping DDM; and iii) adaptivity algorithms in 3D. In this work, these three building blocks have been successfully introduced in a FEM code with a systematic procedure based on the method of manufactured solutions (MMS). Moreover, a three-level parallelization (at the algorithm level, with DDM; at the process level, with message passing interface (MPI), and at the thread level, with OpenMP) has been developed using the paradigm of modular programming which eases the software maintenance and the introduction of new features. Regarding first building block, a family of basis functions which follows a sound mathematical approach to expand the correct space of functions is developed and particularized for triangular prisms. Also, to ease the introduction of different approximation orders in the same mesh, hierarchical basis functions from other authors are used as a black box. With respect to DDM, a thorough study of the error introduced by the non-conformal interfaces between subdomains is required for the adaptivity algorithm. Thus, a quantitative analysis is detailed including non-conformalities generated by independent refinements in neighbor subdomains. This error has not been assessed with detail in the literature and it is a key factor for the adaptivity algorithm to perform properly. An adaptivity algorithm with triangular prisms is also developed and special considerations for the implementation are explained. Finally, on top of these three building blocks, the proof of concept of adaptivity with DDM is discussed.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Daniel Segovia Vargas.- Secretario: David Pardo Zubiaur.- Vocal: Romanus Dyczij-Edlinge

Randomized Computations for Efficient and Robust Finite Element Domain Decomposition Methods in Electromagnetics

Numerical modeling of electromagnetic (EM) phenomenon has proved to become an effective and efficient tool in design and optimization of modern electronic devices, integrated circuits (IC) and RF systems. However the generality, efficiency and reliability/resilience of the computational EM solver is often criticised due to the fact that the underlying characteristics of the simulated problems are usually different, which makes the development of a general, \u27\u27black-box\u27\u27 EM solver to be a difficult task. In this work, we aim to propose a reliable/resilient, scalable and efficient finite elements based domain decomposition method (FE-DDM) as a general CEM solver to tackle such ultimate CEM problems to some extent. We recognize the rank deficiency property of the Dirichlet-to-Neumann (DtN) operators involved in the previously proposed FETI-2$\lambda$ DDM formulation and apply such principle to improve the computational efficiency and robustness of FETI-2$\lambda$ DDM. Specifically, the rank deficient DtN operator is computed by a randomized computation method that was originally proposed to approximate matrix singular value decomposition (SVD). Numerical results show a up to 35\% run-time and 75% memory saving of the DtN operators computation can be achieved on a realistic example. Later, such rank deficiency principle is incorporated into a new global DDM preconditioner (W-FETI) that is inspired by the matrix Woodbury identity. Numerical study of the eigenspectrum shows the validity of the proposed W-FETI global preconditioner. Several industrial-scaled examples show significant iterative convergence advantage of W-FETI that uses 35%-80% matrix-vector-products (MxVs) than state-of-the-art DDM solvers

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Innovative Approaches to the Numerical Approximation of PDEs

This workshop was about the numerical solution of PDEs for which classical approaches, such as the finite element method, are not well suited or need further (theoretical) underpinnings. A prominent example of PDEs for which classical methods are not well suited are PDEs posed in high space dimensions. New results on low rank tensor approximation for those problems were presented. Other presentations dealt with regularity of PDEs, the numerical solution of PDEs on surfaces, PDEs of fractional order, numerical solvers for PDEs that converge with exponential rates, and the application of deep neural networks for solving PDEs