Time-domain Finite Element Research Articles

A novel TDFEM scheme based on high‐performance parallel solver PaStiX

AbstractIn this paper, we focus on the theory and fulfillment of parallel time domain finite element method (TDFEM) with PaStiX (Parallel Sparse matriX package) for solving three‐dimensional electromagnetic (EM) radiation problems. The motivation behind the research is required to maintain numerical stability and achieve high speed for parallel sparse LU factorization on parallel platforms. Firstly, we overview the concept of an unconditional stable TDFEM scheme, and numerical optimization for ill‐conditioned equations obtained from the TDFEM with perfectly matched layer (PML) was presented to alleviate nonconvergent results. Secondly, we develop sparse linear system solvers that use DS factorization to solve the preconditioned system. Numerical results were obtained from the cases of the cavity resonance problem and a cone antenna. A noteworthy observation from the time‐consuming results in different solvers is that PaStiX is usually much faster compared with UMFPACK and SuperLU.

Stabilization of bianisotropic fluid models for time domain simulation

Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropy [Phys. Rev. B 96, 104303 (2017)]. Recent work has shown that, when wavevectors exceed a critical magnitude, the currently employed model for passive Willis fluids possesses a linear instability stemming from the truncation of higher-order coupling terms in the derivation of the constitutive relationships [J. Acoust. Soc. Am. 144(3), 1832 (2018)]. While this instability can be avoided when using frequency domain methods, it poses significant problems for time domain simulations, as short-wavelength numerical noise may grow exponentially. In the present work, we report on methods of stabilizing the Willis fluid model. Specifically, we consider mathematical manipulations of the wave equation derived from the lowest-order constitutive equations, similar to methods that have been explored in continuum approximations of nonlinear spring-mass chains, as well as higher-order coupling terms. Physical interpretations and implications for time domain finite difference and finite element simulations are discussed. [Work supported by the postdoctoral program at ARL:UT.]Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropy [Phys. Rev. B 96, 104303 (2017)]. Recent work has shown that, when wavevectors exceed a critical magnitude, the currently employed model for passive Willis fluids possesses a linear instability stemming from the truncation of higher-order coupling terms in the derivation of the constitutive relationships [J. Acoust. Soc. Am. 144(3), 1832 (2018)]. While this instability can be avoided when using frequency domain methods, it poses significant problems for time domain simulations, as short-wavelength numerical noise may grow exponentially. In the present work, we report on methods of stabilizing the Willis fluid model. Specifically, we consider mathema...

Wave interaction with nonlinear damage and generation of harmonics in composite structures

This work presents a generic Finite Element (FE) based computational scheme for quantifying guided wave interaction with Localised Nonlinear Structural Damage (LNSD) within structures of arbitrary layering and geometric complexity. The through-thickness mode-shape of the structure is obtained through a wave and finite element method. This is applied in a time domain FE simulation in order to generate time harmonic excitation for a specific wave mode. Interaction of the wave with LNSD within the system is computed through an element activation and deactivation iteration. The scheme is validated against experimental measurements and a WFE-FE methodology for calculating wave interaction with damage. Case studies for guided wave interaction with crack and delamination are presented to exhibit the robustness of the proposed method in classifying and identifying damage.

Effect of Variable Thermal Conductivity on the Generalized Thermoelasticity Problems in a Fiber-Reinforced Anisotropic Half-Space

Fiber-reinforced materials have widespread applications, which prompt the study of the effect of fiber reinforcement. Research studies have indicated that thermal conductivity cannot be considered as a constant, which is closely related to temperature change. Based on those studies, we investigate the fiber-reinforced generalized thermoelasticity problem under thermal stress, with the consideration of the effect of temperature-dependent variable thermal conductivity. The problem is assessed according to the L-S theory. A fiber-reinforced anisotropic half-space is selected as the research model, and a region of its surface is subjected to a transient thermal shock. The time-domain finite element method is applied to analyze the nonlinear problem and derives the governing equations. The nondimensional displacement, stress, and temperature of the material are obtained and illustrated graphically. The numerical results reveal that the variable conductivity significantly influences the distribution of the field quantities under the fiber-reinforced effect. And also, the boundary point of thermal shock is the most affected. The obtained results in this paper can be applied to design the fiber-reinforced anisotropic composites under thermal load to satisfy some particular engineering requirements.

Piecewise linear recursive convolution finite element time domain method for electromagnetic analysis of dispersive media

The constitutive relation of dispersive media between displacement vector and electric field in time domain is complicated convolution which is hardly to calculate directly. In this paper, based on finite-element time-domain, combined total-field and scattered-field decomposition technique, the weak form solution equations including electric field E and convolution function ψ for dispersive media are described, and then the plane wave in total-field region containing dispersive media is introduced. The iterative formulas of piecewise linear recursive convolution finite element time domain dealing with constitutive relation for dispersive media are derived. The far scattering field calculation of piecewise-linear recursive convolution for three kinds of general dispersive media models, i.e. Drude model, Debye model and Lorentz model is implemented.

Research and application of the FETD method for the EM transient response of MTLs excited by HEMP

An efficient field-to-line coupling model in the time domain is developed combining the second-order decoupled multiconductor transmission lines (MTLs) equations and the finite-element time-domain (FETD) method for evaluating the electromagnetic (EM) transient response of MTLs over a lossy ground excited by a high-altitude EM pulse (HEMP). First, the incident HEMP electric fields on the lines are calculated as equivalent distributed sources introduced into the MTLs equations. Then, the discrete FETD equations for the lines against the HEMP, in which the effect of frequency-dependent parameters caused by the ground impedance is considered, are derived using the FETD method with current as the single iteration variable. Finally, the transient currents induced on the lines are obtained by assembling all the discrete FETD equations into the global FETD equation, and the transient induced voltages on the lines are calculated by discretising the coupled equations with respect to current and voltage. The proposed model is successfully compared with the finite-difference time-domain method via several examples. It is then employed to analyse the effect of various parameters on the transient currents including the polarisation and the incident angles of the HEMP, the conductivity, and the relative permittivity of the soil.

3-D Implicit–Explicit Hybrid Finite Difference/Spectral Element/Finite Element Time Domain Method Without a Buffer Zone

A nonconformal hybrid finite difference time domain (FDTD)/finite element time domain (FETD) method was previously introduced, which implemented the hybridization through a buffer zone. Although this method has been demonstrated to be accurate and long-time stable, further efforts are still desirable to remove the buffer zone and to implement an implicit–explicit time integration from the perspective of practical applications. In this paper, a novel hybrid method is proposed, which not only successfully eliminates the necessity of the buffer zone without compromising the featured advantage (e.g., nonconformal mesh) but also effectively applies an implicit–explicit time integration scheme to improve the computational efficiency. Furthermore, the new method extends the hybridization to a broader level by incorporating the spectral element time domain (SETD) method based on the discontinuous Galerkin and domain decomposition techniques, resulting in a more general hybrid FDTD/SETD/FETD framework. The framework employs the explicit leapfrog time integration for the FDTD region while it employs the implicit Crank–Nicolson time integration for the FETD region. For the SETD region, either the implicit or explicit time integration can be employed, depending on the mesh sizes in it. When the implicit region becomes large, it can be further split into multiple subdomains to reduce computational complexity. Numerical examples are included to demonstrate the performance of the proposed hybrid method, which is accurate, long-time stable and more efficient than the hybrid method with a buffer zone.

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.

Local Joule heating and electric force on biological membrane during electro-microinjection

Microinjection is one of the key technologies to introduce foreign molecules into the animal cell and embryos. Recently, a novel microinjection method for piercing relatively large egg (∼ϕ1 mm) utilizing the short electric pulse to pierce the biomembrane was developed. Although this method, electro-microinjection, was proved to be capable to pierce the stiff shell of fish egg without fatal damage, its mechanism has not been understood enough to optimize the parameter and the configuration of electro-microinjection. In this study, we have clarified the mechanism of electro-microinjection by numerically calculating the electric and thermal fields during electro-microinjection. A numerical model based on the time domain finite element method (TDFEM) with the measured reliable electric properties of fish egg was developed to calculate the intra-cell electric and thermal fields in detail. The calculation was performed in the conditions of minimal required pulse strength for piercing (227–1470 V) at various given pulse length (10 μs–1 ms) that were obtained from our previous research. The results revealed that the applied pulse generates a locally focused high temperature near the needle electrode and an electric force of 80 μN at its maximum which draws the fish egg to the injection needle. It was found that at least the local temperature of 330 K near the tip of piercing needle by Joule heating is required for electro-microinjection to decrease the rupture stress of the stiff biological shell of fish egg.

A Perfectly Matched Layer for the Nonlinear Dispersive Finite-Element Time-Domain Method

A novel implementation of a perfectly matched layer (PML) is presented for the truncation of finite-element time-domain (FETD) meshes containing electrically complex materials, exhibiting any combination of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Based on the complex coordinate stretching formulation of the PML, the presented technique yields an artificial absorbing layer whose matching condition is independent of material parameters. Moreover, by virtue of only modifying spatial derivatives, the incorporation of the PML into existing solvers for complex media is simple and straightforward. The resulting material-independent PML is incorporated into a nonlinear dispersive method for the vector wave equation, which leverages the $z$ transform and Newton-Raphson techniques to yield an implementation free from recursive convolutions, auxiliary differential equations, and linearizations. This permits the unprecedented truncation and attenuation of nonlinear phenomena, such as spatial and temporal solitons, within the FETD method.

Improving reconstruction of tunnel lining defects from ground-penetrating radar profiles by multi-scale inversion and bi-parametric full-waveform inversion

Complex irregular defects of tunnel linings under complicated geological conditions cannot be accurately reconstructed with the traditional full-waveform inversion (FWI) method due to their irregular geometrical characteristics and complex dielectric properties. Because of extensive inversion calculations and high memory requirements, the traditional FWI method is very sensitive to the initial model and plunge into a local minimum or cycle skipping. To solve this problem, a novel ground-penetrating radar (GPR) FWI method involving two parameters (i.e. permittivity and conductivity) is proposed for improving reconstruction accuracy of lining defects using the total variation (TV) regularization. First, the Delaunay unstructured triangular mesh in finite element time-domain (FETD) method is employed to perform GPR forward modeling, and then the total-variation model constraint and multi-scale inversion strategy are implemented during execution of the conjugate gradient (CG) algorithm, which facilitates the quick search for the global optimal minimum value, thus guaranteeing the avoidance of the ill-posed problem during the inversion process. Accordingly, the detailed features of lining defects can be characterized and reconstructed even for those complicated geological conditions, more specifically, the results show that, with fewer iterations (up to 59 times less), the proposed method present a lower reconstruction error both on permittivity (up to 12.05% lower) and conductivity (up to 7.35% lower). From a comparison of the inversion results and the model, it can be concluded that the proposed FWI algorithm can effectively eliminate the non-physical oscillation and artifacts in the image reconstruction, which may significantly improving the accuracy of defects interpretation and assessing the severity of complex defects.

Through-Wall Detection of the Moving Paths and Vital Signs of Human Beings

Detection of human activities in complex environments such as through wall by ultrawideband radar has many important applications in security, vital rescue, and so on. It is much more difficult to detect vital signs of moving human beings than static ones. In this letter, we build a model for moving targets and apply the time domain finite element method to simulate single-input multiple-outputs (SIMO) radar data. Human respiration is modeled by changing body size and physical parameters. The background removal is performed for radar data. Then, we use the back projection to reconstruct the consecutive target locations, which constitute the moving path, leading to a curve carrying vital signs in the radar image. Since SIMO radar data are multivariate, we use multivariate empirical mode decomposition (MEMD) and fast Fourier transform to separate and extract the respiratory characteristic frequencies. The reconstructed frequency coincides with that in the original model. The result shows that the combination of SIMO radar and MEMD can effectively identify the moving path of the human being behind the wall and extract vital signs.

An uncoupled numerical approach for soil consolidation analysis

This work proposes a simple new “uncoupled” numerical procedure to analyze the static behavior of saturated porous media, taking into account standard time-domain finite element discretizations. In this novel approach, each phase of the coupled problem is treated separately, providing simpler, smaller and better conditioned systems of equations, which greatly reduces the solving effort of the pore-model and enables a more effective solution technique. In the proposed new algorithm, a novel spatially discretized matrix is introduced into the analysis, and its (eventual) approximating errors are properly counterbalanced at consecutive steps, greatly improving the accuracy of the uncoupled solution. At the end of the paper, numerical examples are presented, illustrating the performance of the discussed methodologies.

An Efficient Mixed Finite-Element Time-Domain Method for Complex Electrically Small Problems

A new mixed finite-element time-domain (FETD) method is developed to eliminate the low-frequency breakdown phenomenon in the conventional finite-element method when solving electrically small transient problems. Constructed by Gauss’s law and the current continuity equation, the divergence constraint equation is incorporated into the wave equation for the electric field in the form of a Lagrange multiplier. Curl-conforming vector basis functions and nodal basis functions are chosen for spatial discretization and the implicit Newmark-beta algorithm is adopted for time integration. It is shown that the terms constituted by the divergence constraint equation in the system matrix help to suppress the effect of the singularity of the stiffness matrix, and the property of the system matrix is significantly improved. Numerical results show that for electrically small problems that cannot be solved by the conventional FETD method, the proposed method not only obtains a stable numerical solution but also achieves a faster convergence rate when an iterative solver is used, thus improving the computational efficiency.

Auxiliary differential equation finite-element time-domain method for electromagnetic analysis of dispersive media

A new technique of Finite-Element Time-Domain (FETD) for electromagnetic analysis of dispersive media is presented. Firstly, the plane wave and matrix equation in dispersive media is obtained. An iteration equation contains Jp and E is presented. Followed by an introduction of auxiliary differential Equation to the FETD method, a recursive formulation of constitutive for Jp and E in time domain is given. Then the achievement of Auxiliary Differential Equation Finite-Element Time-Domain (ADE-FETD) is developed through combining the above two equations. Finally, the feasibility of this technique is validated with three-dimension numerical examples.

Numerical simulation of falling/heavy weight deflectometer test considering linear viscoelastic behaviour in bituminous layers and inertia effects

Falling (or Heavy) Weight Deflectometer (F/HWD) is a widely used non-destructive device for pavement evaluation. The falling mass system of this device can generate a load representative of a wheel. Deflections are monitored by geophones at several distances from the loading centre. Interpretation of F/HWD deflections is classically done using a static elastic method. However, the F/HWD test is clearly dynamic and the mechanical behaviour of bituminous materials is most likely linear viscoelastic at the temperature and strain conditions of the test. This paper aims at exploring the differences due to the consideration of dynamic and viscoelastic effects of the pavement structure in the numerical simulation of the F/HWD test. Deflections of an existing pavement structure were numerically simulated through a time-domain finite element method. Four numerical simulations are proposed: static elastic, dynamic elastic, static viscoelastic and dynamic viscoelastic. This study shows that dynamic and viscoelastic effects significantly affect time-history and peak values of simulated deflections.

Predicting absorption characteristics of single-leaf permeable membrane absorbers using finite element method in a time domain

Permeable membranes (PMs), which are air-permeable thin woven fabrics or non-woven fabrics, are attractive sound-absorbing materials used to control acoustics in buildings. Although the absorption characteristics of various PM absorbers have been studied extensively, it has not been established how to incorporate PMs into a wave-based acoustic simulation method known as the time-domain finite element method (TD-FEM). This paper presents a numerical model of limp PMs for TD-FEM as well as a computationally efficient TD-FEM for predicting sound fields including PMs. The limp PM model incorporates effects of both sound-induced vibration of PMs and the air permeability of PMs. Consequently, two material parameters are considered: the surface density and flow resistance of PMs. Verification of the limp PM model and efficiency evaluation of TD-FEM were first performed with an impedance tube problem. Then, reverberation absorption coefficients of single-leaf PM absorbers with a non-locally reacting rigid backed air cavity were predicted through simulation of a reverberation absorption coefficient measurement using the present TD-FEM at frequencies of 100 Hz to 2.5 kHz. The predicted results were compared with measured and theoretical values. To demonstrate the applicability of present TD-FEM, eight PMs, each with different flow resistance and surface density, were considered for comparison. Results revealed that the present TD-FEM with the limp PM model can predict the magnitude relation of absorption characteristics of single-leaf PM absorbers attributable to differences of material properties of PMs.

A block rational Krylov method for 3-D time-domain marine controlled-source electromagnetic modelling

We introduce a novel block rational Krylov method to accelerate three-dimensional time-domain marine controlled-source electromagnetic modeling with multiple sources. This method approximates the time-varying electric solutions explicitly in terms of matrix exponential functions. A main attraction is that no time stepping is required, while most of the computational costs are concentrated in constructing a rational Krylov basis. We optimize the shift parameters defining the rational Krylov space with a positive exponential weight function, thereby producing smaller approximation errors at later times and reducing iteration numbers. Furthermore, for multi-source modeling problems, we adopt block Krylov techniques to incorporate all source vectors in a single approximation space. The method is tested on two examples: a layered seafloor model and a 3D hydrocarbon reservoir model with seafloor bathymetry. The modeling results are found to agree very well with those from 1D semi-analytic solutions and finite-element time-domain solutions using a backward Euler scheme, respectively. Benchmarks of the block rational Krylov method demonstrate that it can be as up to 10 times faster than backward Euler. The block method also benefits from better memory efficiency, resulting in considerable speedup compared to non-block methods.

A partially staggered discontinuous Galerkin method for transient electromagnetics

Conservation of energy, spurious-free, and optimal convergence are among the most favorite ingredients of a discontinuous Galerkin finite-element time-domain method (DG-FETD) for the analysis of transient electromagnetic problems. Unfortunately, due to the reliance on dissipative upwind flux to suppress spurious modes, most existing DG-FETDs possess only one or two of them. In this paper, we present a novel two-dimensional DG-FETD method, which combines all of the three advantages. The proposed method is based on the electric field E and magnetic field H, whose approximation spaces are composed of Whitney's edge functions. To inhibit the spurious modes while keeping the conservation of numerical energy, the numerical flux with dissipation is not adopted. Instead, a carefully designed interface condition along with the non-dissipative central flux is employed for this purpose. That is, across each face of an element except for those coinciding with boundaries of the computational domain, if E is strongly enforced to be tangential continuous then the continuity of H is weakly imposed, and vice versa. For the convenience of complying with this requirement and without loss of the inherent parallelism of the DG-FETD method, two partially staggered sets of subdomains are built for E and H, respectively. These subdomains are made up of sub-elements generated through a refinement of each cell on the initial mesh into four similar elements. The final discrete system is advanced with a second order leap-frog time-stepping scheme. A series of numerical examples demonstrate that the present method is robust and much more superior to the conventional DG-FETD methods.

A 2.5D time-frequency domain model for railway induced soil-building vibration due to railway defects

A new hybrid time-frequency modelling methodology is proposed to simulate the generation of railway vibration caused by singular defects (e.g. joints, switches, crossings), and its propagation through the track, soil and into nearby buildings. To create the full source-to-received model, first the force density due to wheel-rail-defect interaction is calculated using a time domain finite element vehicle-track-soil model. Next, the frequency domain track-soil transfer function is calculated using a 2.5D boundary/finite element approach and coupled with the force densities to recover the free-field response. Finally, the soil-structure interaction of buildings close to the line is computed using a time domain approach. The effect of defect type, train speed and building type (4-storey office block and 8-storey apartment building) on a variety of commonly used international vibration metrics (one-third octaves, PPV, MTVV) is then investigated. It is found that train speed doesn’t correlate with building vibration and different defect types have a complex relationship with vibration levels both in the ground and buildings. The 8-storey apartment building has a frequency response dominated by a narrow frequency range, whereas the modal contribution of the 4-storey office building is over a wider frequency band. This results in the 8-storey building having a higher response.