Finite-element Time-domain Algorithm Research Articles

A Parallel Finite-Element Time-Domain Method for Nonlinear Dispersive Media

In this article, a novel use of graphics processing units (GPUs) is presented for the acceleration of finite-element time-domain (FETD) methods containing electrically complex media. By leveraging the massively parallel architecture of the GPU via NVIDIA's Compute Unified Device Architecture (CUDA) language, the immense computational burden imposed by these materials can be largely alleviated, facilitating their modeling and incorporation into electromagnetic devices and systems. To that end, an analysis of both mixed and vector wave equation-based nonlinear dispersive FETD algorithms is presented in order to both identify computational bottlenecks and determine their amenability to parallelization. Based on this analysis, a parallel elemental matrix-evaluation procedure is proposed, which when coupled to the recently derived Gaussian belief propagation method for matrix assembly and solution, demonstrates a performance increase of up to 200 times as compared with a traditionally serial implementation.

Simulation of ground penetrating radar on dispersive media by a finite element time domain algorithm

This paper presents a two-dimensional finite element time domain (FETD) algorithm for simulation of ground penetrating radar (GPR) on dispersive and lossy media. The medium of linear dispersion is modeled by a multi-pole Debye model, which is incorporated into the FETD scheme through the Fourier transform. We apply the finite element Galerkin method to discretize the computation domain using triangular elements. A perfectly matched layer is extended to match the dispersive media and is used as an absorbing boundary condition to truncate the computation domain. The proposed FETD algorithm is proven to have comparable accuracy with the finite difference time domain (FDTD) method by comparing the simulated electromagnetic waves propagating in a homogeneous dispersive medium with the corresponding analytical solution. At an expense of computational costs, the FETD approach can yield more accurate results when simulating a complex geometry than the FDTD approach, which suffers from the staircase error. Moreover, we analyze the influence of material dispersion on GPR data acquired by different survey geometries. The results demonstrate that material dispersion will bring a great challenge for quantitative interpretation of GPR data. Numerical simulation of GPR data with dispersion does help in accurate interpretation of GPR data.

Specific evaluation of tunnel lining multi-defects by all-refined GPR simulation method using hybrid algorithm of FETD and FDTD

Irregular and complicated tunnel lining defects cannot be finely depicted with the standard finite difference simulation method based on regular mesh due to its irregular geometrical characteristics and complex physical properties of various defects. To solve this problem, a new hybrid algorithm is proposed for all-refined ground penetrating radar (GPR) simulation by combining the finite element time domain (FETD) and finite difference time domain (FDTD), which employs FETD algorithm in main region based on unstructured Delaunay mesh and Newmark-β implicit FDTD algorithm in PML region based on convolutional perfectly matched layer (CPML), respectively. The calculation efficiency and simulation accuracy of different algorithms methods are compared in detail using a representative numerical model with a water-filled void embedded. In addition, several complicated numerical models involving different typical defects and mixed case (e.g. non-compactness, voids, leaking with fractured and cracks, etc.) are also conducted, the results presented show that the proposed hybrid algorithm helps GPR practitioners to gain a deeper insight into the effects of multi-defects on the propagation characteristic of electromagnetic wave, and provides more precise technical guidance for defect interpretation. Moreover, spectrum analysis of echoes responses from defects, by using the short-time Fourier transform (STFT), is also presented, including amplitude spectrum, frequency spectrum and mode of dominant frequency.

Dispersive Möbius Transform Finite-Element Time-Domain Method on Graphics Processing Units

A novel use of graphics processing units (GPUs) is presented in the execution of the dispersive finite-element time-domain (FETD) method, based upon the Mobius (bilinear) ${z}$ -transform technique. By utilizing the immense computational power of modern GPUs via NVIDIA’s compute unified device architecture (CUDA) language, a narrowing of the performance gap, which currently exists between dispersive FETD methods and their non-dispersive counterparts, can be achieved, thus facilitating the study of a wider range of physical phenomena. An analysis of the $ {z}$ -transform dispersive FETD algorithm is presented in order to both identify dispersive overhead bottlenecks and determine its suitability to parallelization. Numerical studies are then undertaken to measure the performance increase as a function of simulation parameters, such as number of variables, the amount of dispersive material present, and floating point precision.

Analysis of the electromagnetic response of complex dispersive material with the single-step wideband finite element method

Abstract The paper describes the formulation and implementation of the broadband finite element time domain algorithm. The presented formalism is valid to analysis of electromagnetic phenomena in linear, frequency selective materials. The complex profile of permittivity of materials is approximated using a set of the Lorentz resonance models. The solution of the integro-differential second order equation is obtained using a singlestep integration scheme and a recursive convolution algorithm. The discussed formulation enables to adopt the structure of the narrowband part as well as the phase of calculation of the convolution equations for the subsequent components. The properties of the algorithm are validated using a finite difference broadband algorithm.

3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach

We present a finite-element time-domain (FETD) approach for the simulation of 3D electromagnetic (EM) diffusion phenomena. The finite-element algorithm efficiently simulates transient electric fields and the time derivatives of magnetic fields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms. To compute transient electromagnetic fields, the electric field diffusion equation is transformed into a system of differential equations via Galerkin’s method with homogeneous Dirichlet boundary conditions. To ensure numerical stability and an efficient time step, the system of the differential equations is discretized in time using an implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver along with a fill-in reduced ordering technique. When advancing the solution in time, the FETD algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution. To simulate a step-off source waveform, the 3D FETD algorithm also incorporates a 3D finite-element direct current (FEDC) algorithm that solves Poisson’s equation using a secondary potential method for a general anisotropic earth model. Examples of controlled-source FETD simulations are compared with analytic and/or 3D finite-difference time-domain solutions and are used to confirm the accuracy and efficiency of the 3D FETD algorithm.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

Mixed Finite-Element Time-Domain Method for Transient Maxwell Equations in Doubly Dispersive Media

We describe a mixed finite-element time-domain algorithm to solve transient Maxwell equations in inhomogeneous and doubly dispersive linear media where both the permittivity and permeability are functions of frequency. The mixed finite-element time-domain algorithm is based on the simultaneous use of both electric and magnetic field as state variables with a mix of edge (Whitney 1-form) and face (Whitney 2-form) elements for discretization of the coupled first-order Maxwell curl equations. The constitutive relations are decoupled from the curl equations and cast in terms of (auxiliary) ordinary differential equations involving time derivatives. Permittivity and permeability dispersion models considered here are quite general and recover Lorentz, Debye, and Drude models as special cases. The present finite-element time-domain algorithm also incorporates the perfectly matched layer absorbing boundary conditions in a natural way.

Analysis and Simulation of Rotating Magnetic Field Diffusion Through a Parallelogram Hysteresis Model

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper deals with the diffusion of rotating electromagnetic fields into hysteretic materials. A simplified vector formulation based on a parallelogram hysteresis model has been investigated, in order to significantly reduce the computational effort. Vector saturation and loss properties are discussed. The whole simulation has been done by the use of a finite-element time-domain algorithm. </para>

On the Investigation of Phase Error of Cubic Wavelet-Like Functions in a Finite Element Time Domain Algorithm

In this research, the phase error associated with using cubic wavelet-like functions in a finite element time domain solution of Maxwell's equations in two dimensions is investigated. A brief discussion of the generation of the cubic wavelet-like functions is presented. Several example problems will be presented in which some of the advantages and disadvantages of the algorithm will be discussed. In these examples, solutions obtained with the finite element time domain algorithm utilizing cubic wavelet-like functions, analytic solutions, and solutions obtained by the traditional finite difference time domain method will be compared.

Performance of preconditioners for the distributed vector finite-element time-domain algorithm

This paper deals with some aspects of performance of the symmetric successive over-relaxation preconditioner in a distributed environment. The details of distributed formulation of the preconditioner are presented. Some performance metrics are compared and discussed for the message passing interface implementation of the algorithm. The properties of the solver are estimated for concurrent three-dimensional formulation of the finite-element time-domain method. The analyzed benchmark models are approximated by tetrahedral first order Whitney elements.

Stability and Fidelity of the Finite Element Time Domain Method With Distorted Mesh

Properties of the three-dimensional formulation of the finite element time domain algorithm for the wave equation are analyzed. First-order edge elements are implemented in the formulation. Several issues associated with deformation of the structured mesh and efficiency of the time integration scheme are presented. The convergence and stability of the time domain algorithm depending on the spatial discretization are discussed. The numerical accuracy of the simulation is studied.

Numerical Performance of the Distributed Vector Finite-Element Time-Domain Algorithm

This paper deals with a distributed time-domain modeling of electromagnetic phenomena with the finite-element method. The model is approximated by edge elements. The constitutive equations and method of parallelization of the algorithm are presented. The properties of the distributed finite-element time-domain algorithm are discussed. Some typical performance metrics are studied for the parallel versions of the software. The presented algorithm is executed on a heterogeneous and a homogeneous clusters of workstations. Two different distributed memory environments (MPI and PVM) are used to evaluate the efficiency of the algorithm.

An unconditionally stable extended (USE) finite-element time-domain solution of active nonlinear microwave circuits using perfectly matched layers

This paper proposes an extension of the unconditionally stable finite-element time-domain (FETD) method for the global electromagnetic analysis of active microwave circuits. This formulation has two advantages. First, the time-step size is no longer governed by the spatial discretization of the mesh, but rather by the Nyquist sampling criterion. Second, the implementation of the truncation by the perfectly matched layers (PML) is straightforward. An anisotropic PML absorbing material is presented for the truncation of FETD lattices. Reflection less than -50 dB is obtained numerically over the entire propagation bandwidth in waveguide and microstrip line. A benchmark test on a microwave amplifier indicates that this extended FETD algorithm is not only superior to finite-difference time-domain-based algorithm in mesh flexibility and simulation accuracy, but also reduces computation time dramatically.

An efficient implementation of the finite-element time-domain algorithm on parallel computers using a finite-element tearing and interconnecting algorithm

An efficient algorithm for implementing the finite-element time-domain (FETD) method on parallel computers is presented. An unconditionally stable implicit FETD algorithm is combined with the finite-element tearing and interconnecting (FETI) method. This domain decomposition algorithm converges at a rate dependent solely on the subdomain size, resulting in a highly scalable algorithm. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 16:204–208, 1997.

A comparison of the performance of the finite difference time‐domain, finite element time‐domain, and planar generalized Yee algorithms of high‐performance parallel computers

AbstractParallel algorithms for the finite difference time‐domain (FDTD), the planar generalized Yee (PGY), and the finite element time‐domain (FETD) methods are presented. The FDTD and the PGY algorithms are both explicit time‐domain solutions of Maxwell's equations, while the PGY algorithm is based on an unstructured grid. The FETD algorithm is a semi‐implicit solution of Maxwell's equations using variational principles, and thus requires a matrix inversion for every time iteration. The three parallel algorithms are based on spatial decompositions of the discrete three‐dimensional problem spaces. A comparative analysis of the parallel algorithms is presented based on their memory and computational efficiency as well as their parallel efficiency.