Mesh Truncation Research Articles

Leaky mode analysis using complex infinite elements

We propose a novel numerically efficient model for accurate characterization of leaky modes in optical waveguides. Within the context of finite elements, the mesh boundaries are truncated using complex infinite elements (CIEs). The CIE is based on adopting the homogeneous solution of the wave equation with the shape function of infinite elements to accurately and physically model semi-infinite subdomains instead of placing an artificial layer such as PML. The proposed treatment can be easily implemented and requires less computational resources compared to other conventional mesh truncation methods. The accuracy and rigor of our approach are demonstrated through studying different leaky waveguides’ configurations.

Improvement of selection formulas of mesh size and truncation numbers for the DE-Sinc approximation and its theoretical error bound

Improvement of selection formulas of mesh size and truncation numbers for the DE-Sinc approximation and its theoretical error bound

Parametrization-free locally-conformal perfectly matched layer method for finite element solution of Helmholtz equation

We present a novel Locally-Conformal Perfectly Matched Layer (LCPML) method for mesh truncation in the Finite Element Method (FEM), to solve wave equations derived from Maxwell's equations. The original LCPML method was proposed by Ozgun and Kuzuoglu in 2006 [16] by utilizing the concept of complex elements in which nodal coordinates of elements are replaced by complex coordinates obtained by using a complex coordinate-transformation. In the proposed LCPML method, named as LCPML-log method, a more efficient coordinate-transformation is used together with new FEM formulation and implementation. The proposed method offers the advantage that only a few (or even a single) PML layer is sufficient to get reliable results. In addition, it does not require any PML parameter tuning, and hence, the decaying wave behavior in PML is achieved self-adaptively. The numerical results of some detailed and systematic performance analyses are also presented.

Optimal selection formulas of mesh size and truncation numbers for the double-exponential formula

Optimal selection formulas of mesh size and truncation numbers for the double-exponential formula

Strategies to parallelize a finite element mesh truncation technique on multi-core and many-core architectures

Achieving maximum parallel performance on multi-core CPUs and many-core GPUs is a challenging task depending on multiple factors. These include, for example, the number and granularity of the computations or the use of the memories of the devices. In this paper, we assess those factors by evaluating and comparing different parallelizations of the same problem on a multiprocessor containing a CPU with 40 cores and four P100 GPUs with Pascal architecture. We use, as study case, the convolutional operation behind a non-standard finite element mesh truncation technique in the context of open region electromagnetic wave propagation problems. A total of six parallel algorithms implemented using OpenMP and CUDA have been used to carry out the comparison by leveraging the same levels of parallelism on both types of platforms. Three of the algorithms are presented for the first time in this paper, including a multi-GPU method, and two others are improved versions of algorithms previously developed by some of the authors. This paper presents a thorough experimental evaluation of the parallel algorithms on a radar cross-sectional prediction problem. Results show that performance obtained on the GPU clearly overcomes those obtained in the CPU, much more so if we use multiple GPUs to distribute both data and computations. Accelerations close to 30 have been obtained on the CPU, while with the multi-GPU version accelerations larger than 250 have been achieved.

An Efficient Parallel Hybrid Method of FEM-MLFMA for Electromagnetic Radiation and Scattering Analysis of Separated Objects

In order to meet the rapidly increasing demand for accurate and efficient analysis of complex radiating or scattering structures in the presence of electrically large objects, a finite element method (FEM)-multilevel fast multipole algorithm (MLFMA) hybrid method that based on the Finite Element-Iterative Integral Equation Evaluation (FE-IIEE) mesh truncation technique is proposed in this paper. The present method makes use of FEM for the regions with small and complex features and MLFMA for the analysis of the electrically large objects, which ensure the accuracy and applicability of the method are better than most commonly adopted FEM-high frequency technique (HFT) hybrid method. The mutual interactions between regions are taken into account in a fully coupled way through iterative near filed computation process. In order to achieve an excellent performance, both algorithms have been implemented together from scratch, being able to run over multi CPU cores. An efficient parallel FEM domain decomposition method (DDM) solver with exploiting geometrical repetitions is included to drastically reduce memory requirements and computational time in the calculation of large array antenna. Also, the parallel MLFMA is adopted to expedite the near-field information exchange between regions. Through numerical example, the effect of distance between regions on the convergence of the proposed hybrid method is studied, and it is shown that the proposed method converge well even if the distance is equal to 0.05λ. Through comparisons with an in-house higher order method of moments (HOMoM) code and commercial software FEKO, the accuracy and effectiveness of the implemented parallel hybrid method are validated showing excellent performance.

Improvement of selection formulas of mesh size and truncation numbers for the double-exponential formula

Improvement of selection formulas of mesh size and truncation numbers for the double-exponential formula

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.

Computational Method for Optimal Guidance and Control Using Adaptive Gaussian Quadrature Collocation

A method is described for computational optimal guidance and control using adaptive Gaussian quadrature collocation and sparse nonlinear programming. The method employs adaptive Legendre–Gauss–Radau (LGR) quadrature collocation using a mesh truncation and remapping procedure at the start of each guidance cycle, thereby retaining only the mesh points associated with the unexpired horizon. Using this approach, the starting mesh for a guidance cycle is smaller than the starting mesh used on the previous guidance cycle. The nonlinear programming problem is then solved on the reduced mesh corresponding to the unexpired horizon to generate the control that is used on the current guidance cycle. It is noted that the mesh points on the unexpired horizon are well placed for rapidly solving the LGR quadrature collocation approximation of the optimal control problem. The method developed in this paper is demonstrated on two challenging aerospace optimal control problems. For both examples, the dynamics are simulated on each guidance cycle using a perturbed dynamic model in the absence and presence of a computation time delay. The results of this study demonstrate that the method developed in this paper is viable as a computational method for optimal guidance and control.

On the use of many-core machines for the acceleration of a mesh truncation technique for FEM

Finite element method (FEM) has been used for years for radiation problems in the field of electromagnetism. To tackle problems of this kind, mesh truncation techniques are required, which may lead to the use of high computational resources. In fact, electrically large radiation problems can only be tackled using massively parallel computational resources. Different types of multi-core machines are commonly employed in diverse fields of science for accelerating a number of applications. However, properly managing their computational resources becomes a very challenging task. On the one hand, we present a hybrid message passing interface + OpenMP-based acceleration of a mesh truncation technique included in a FEM code for electromagnetism in a high-performance computing cluster equipped with 140 compute nodes. Results show that we obtain about 85% of the theoretical maximum speedup of the machine. On the other hand, a graphics processing unit has been used to accelerate one of the parts that presents high fine-grain parallelism.

Simulation of Challenging Electromagnetic Problems Using a Massively Parallel Finite Element Method Solver

This communication presents an efficient massively parallel finite element method solver for the solution of complex and electrically large electromagnetic problems with arbitrary structures. The solver makes use of a domain decomposition algorithm to decompose the original problem into several non-overlapping sub-domains that may be solved independently in parallel through the application of the corresponding transmission conditions on the interfaces of the adjacent sub-domains. A numerical exact mesh truncation algorithm called finite element-iterative integral equation evaluation, accelerated with multilevel fast multipole algorithm, is implemented to meet the highly accurate requirements of today's challenging simulations. What's more, a hybrid message passing interface and an open multi-processing parallel framework are designed to achieve large-scale parallel performance on supercomputers. Through several numerical examples, the accuracy, effectiveness, and scalability of the proposed solver will be demonstrated, achieving more than 60% parallel efficiency on an eight times CPU core scale (from 1280 to 10 240 cores).

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.

Higher-Order Finite Element Electromagnetics Code for HPC environments

In this communication, an electromagnetic software developed to work in High-Performance Computing (HPC) environments is presented. The software is mainly based on the Finite Element Method (FEM) making use of a number of numerical techniques developed by its authors including its own family of higher-order curl-conforming elements and a non-standard mesh truncation methodology for the analysis of open region problems such as those of scattering and radiation. The code is written in FORTRAN 2003 and it adopts from scratch parallel programming paradigms such as Message Passing Interface (MPI) and Open Multi-Processing (OpenMP). It also includes an in-house development to ease the use of remote computer systems and HPC environments. Initially designed for single-user multicore machines and small cluster environments, a number of modifications have been included in the code in order to make it capable of running on large-scale computer systems and hence, be able to deal with larger problems in terms of number of unknowns. Details of the implementation are shown. Numerical results obtained on an HPC system corresponding to the analysis of a few illustrative challenging problems are also shown.

Self-adaptive [formula omitted] finite element method with iterative mesh truncation technique accelerated with Adaptive Cross Approximation

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.

Implementation of coordinate transformations in periodic finite-element method for modeling rough surface scattering problems

The coordinate transformation technique (with its current name of transformation electromagnetics) is applied to the finite-element method (FEM) with periodic boundary conditions for efficient Monte Carlo simulation of one-dimensional random rough surface scattering problems. In a unit cell of periodic structure, two coordinate transformations are used, one of which is a real transformation designed to model the rough surface with flat surface, and the other is a complex transformation used to design a perfectly matched layer for mesh truncation. The approach is easily implementable because only the nodal coordinates are replaced by the transformed coordinates in a standard FEM. Since a simple and single mesh with flat surface is used, Monte Carlo simulations are performed in an efficient manner. Numerical results are presented for various rms surface heights, correlation lengths, and angles of incidence. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:322–329, 2016.

Nonrigorous symmetric second-order ABC applied to large-domain finite element modeling of electromagnetic scatterers

Nonrigorous symmetric second-order absorbing boundary condition (ABC) is presented as a feasible local mesh truncation in the higher-order large-domain finite element method (FEM) for electromagnetic analysis of scatterers in the frequency domain. The ABC is implemented on large generalized curvilinear hexahedral finite elements without imposing normal field continuity and without introducing new variables. As the extension of our previous work, the method is comprehensively evaluated by analyzing several benchmark targets, i.e., a metallic sphere, a dielectric cube, and NASA almond. Numerical examples show that radar cross section (RCS) of analyzed scatterers can be accurately predicted when the divergence term is included in computations nonrigorously. An influence of specific terms in the second-order ABC, which absorb transverse electric (TE) and transverse magnetic (TM) spherical modes, is also investigated. Examples show significant improvements in accuracy of the nonrigorous second-order ABC over the first-order ABC.

Efficient computation of dispersion curves for multilayered waveguides and half-spaces

Motivated by the need to compute dispersion curves for layered media in the contexts of geophysical inversion and nondestructive testing, a novel discretization approach, termed complex-length finite element method (CFEM), is developed and shown to be more efficient than the existing finite element approaches. The new approach is exponentially convergent based on two key features: unconventional stretching of the mesh into complex space and midpoint integration for evaluating the contribution matrices. For modeling the layered half-spaces of infinite depth, we couple CFEM with the method of perfectly matched discrete layers (PMDL) to minimize the errors due to mesh truncation. A number of numerical examples are used to investigate the efficiency of the proposed methods. It is shown that the suggested combination of CFEM and PMDL drastically reduces the number of elements, while requiring minor modifications to the existing finite element codes. It is concluded that the methods’ exponential convergence and sparse computation associated with linear finite elements, result in significant reduction in the overall computational cost.

Higher Order Time-Domain Finite-Element Method for Microwave Device Modeling With Generalized Hexahedral Elements

A novel higher order and large-domain Galerkin-type finite-element method (FEM) is proposed for direct 3-D electromagnetic modeling in the time domain. The method is implemented in the time-domain finite-element method (TDFEM) analysis of multiport microwave waveguide devices with arbitrary metallic and dielectric discontinuities. It is based on the geometrical modeling using Lagrange-type interpolation generalized hexahedra of arbitrary geometrical-mapping orders, field expansion in terms of hierarchical curl-conforming 3-D polynomial vector basis functions of arbitrarily high field-approximation orders, time stepping with an implicit unconditionally stable finite-difference scheme invoking the Newmark-beta method, and mesh truncation introducing the waveguide port boundary condition. Numerical examples demonstrate excellent accuracy, efficiency, stability, convergence, and versatility of the presented method, and very effective large-domain TDFEM models of 3-D waveguide discontinuities using minimal numbers of large conformal finite elements and minimal numbers of unknowns. The results obtained by the higher order TDFEM are in an excellent agreement with indirect solutions obtained from the FEM analysis in the frequency domain (FD) in conjunction with the discrete Fourier transform and its inverse, as well as with measurements and with alternative full-wave numerical solutions in both time and FDs.

Influence of mesh size and truncation order on reconstruction of impact force on composite elastic beams

Abstract Reconstructing impact forces can improve considerably structural health monitoring as the extent of damage can be better made out. In this work an inverse problem formulation to identify the pressure generated by a non punctual impact is investigated. Considering the case of linear elastic layered composite structures, reconstruction of impact pressure is performed through a finite element model of the structure and impulse response functions between the impact zone and sensors placed at known positions. Assuming that the pressure is uniform, reconstruction is carried out by regularized deconvolution based on generalized singular value decomposition. The infl uence of mesh size, modal truncation and time sampling on the reconstructed pressure is discussed.

A fast elasto-plastic formulation with hierarchical matrices and the boundary element method

Boundary element methods offer some advantages for the simulation of tunnel excavation since the radiation condition is implicitly fulfilled and only the excavation and ground surfaces have to be discretized. Hence, large meshes and mesh truncation, as required in the finite element method, are avoided. Recently, capabilities for efficiently dealing with inelastic behavior and ground support have been developed, paving the way for the use of the method to simulate tunneling. However, for large scale three dimensional problems one drawback of the boundary element method becomes prominent: the computational effort increases quadratically with the problem size. To reduce the computational effort several fast methods have been proposed. Here a fast boundary element solution procedure for small strain elasto-plasticity based on a collocation scheme and hierarchical matrices is presented. The method allows the solution of problems with the computational effort and sparse storage increasing almost linearly with respect to the problem size.