Time-domain Boundary Integral Equation Method Research Articles

RI-CDI-FDTD method and program implementation for electromagnetic characteristics simulation of lossy Debye dispersive medium

Dispersive media refer to a class of natural substances, including living organisms, composite materials, plasma and water, with diverse applications in areas such as biomedicine, microwave sensing, electromagnetic protection, and stealth technology. In the pursuit of investigating the electromagnetic properties of these media, time-domain numerical methods, including finite difference in time domain (FDTD), finite element method, and time domain boundary integral equation method, have been widely utilized. Time-domain numerical methods are preferred to their frequency-domain counterparts owing to their ability to handle nonlinear and wideband problems, as well as various material properties. The FDTD method, in particular, is a highly adaptable, robust, and easy-to-use numerical method that directly solves the Maxwell equations while also simulating the reflection, transmission, and scattering of electromagnetic waves in complex dispersion media. Nonetheless, the traditional FDTD method suffers low computational efficiency arising from the Courant-Friedrichs-Lewy (CFL) stability condition. To solve the problem of low computational efficiency, a new method, the complying divergence implicit finite-difference time-domain (CDI-FDTD) method with a one-step leapfrog scheme, is introduced for lossy Debye dispersive media. The Maxwell equations in the frequency domain form a starting point, and the Fourier transform is utilized to transform the electromagnetic field components from the frequency domain to the time domain. To approximate the integral terms arising from the frequency-to-time domain transformation, a recursive integration (RI) method is employed. Subsequently, the time-domain Maxwell equations and auxiliary variables are discretized with a one-step leapfrog implicit scheme. The iterative formula of the RI-CDI-FDTD algorithm for lossy Debye dispersive media is then derived. The RI-CDI-FDTD method does not change the formulas of the traditional CDI-FDTD method while only requiring to add auxiliary variables for updating field components to the dispersive medium region. The numerical implementation is straightforward, and the electromagnetic modeling is flexible. Moreover, the unconditional stability of the RI-CDI-FDTD algorithm is proven by using the von Neumann method. Finally, some numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed method. In conclusion, our work contributes a crucial numerical simulation tool to accurately modeling complex dispersive media while providing a systemic stability analysis method for time-domain numerical methods.

An interpolation-based fast-multipole accelerated boundary integral equation method for the three-dimensional wave equation

A new fast multipole method (FMM) is proposed to accelerate the time-domain boundary integral equation method (TDBIEM) for the three-dimensional wave equation. The proposed algorithm is an enhancem...

An interpolation-based fast-multipole accelerated boundary integral equation method for the three-dimensional wave equation

A new fast multipole method (FMM) is proposed to accelerate the time-domain boundary integral equation method (TDBIEM) for the three-dimensional wave equation. The proposed algorithm is an enhancement of the interpolation-based FMM for the time-domain case, adopting the notion of the plane-wave time-domain algorithm. With the application being targeted at a low-frequency regime, the proposed time-domain interpolation-based FMM can reduce the computational complexity of the TDBIEM from O(Ns2Nt) to O(Ns1+δNt) (where δ=1/3 or 1/2) with the help of multilevel space–time hierarchy, where Ns and Nt are the spatial and temporal degrees of freedom, respectively. The computational accuracy and speed of the proposed accelerated TDBIEM are verified in comparison with those of the conventional (direct) TDBIEM via numerical experiments.

Time-domain simulation for water wave radiation by floating structures (Part A)

Direct time-domain simulation of floating structures has advantages: it can calculate wave pressure fields and forces directly; and it is useful for coupled analysis of floating structures with a mooring system. A time-domain boundary integral equation method is presented to simulate three-dimensional water wave radiation problems. A stable form of the integration free-surface boundary condition (IFBC) is used to update velocity potentials on the free surface. A multi-transmitting formula (MTF) method with an artificial speed is introduced to the artificial radiation boundary (ARB). The method was applied to simulate a semi-spherical liquefied natural gas (LNG) carrier and a semi-submersible undergoing specified harmonic motion. Numerical parameters such as the form of the ARB, and the time and space discretization related to this method are discussed. It was found that a good agreement can be obtained when artificial speed is between 0.6 and 1.6 times the phase velocity of water waves in the MTF method. A simulation can be done for a long period of time by this method without problems of instability, and the method is also accurate and computationally efficient.

A hardware acceleration of the time domain boundary integral equation method for the wave equation in two dimensions

We aimed at reducing the runtime of the time domain boundary integral equation method (TDBIEM) for the two-dimensional scalar wave equation by the help of a special-purpose computer MDGRAPE-2, which is a high-performance vector-parallel computer dedicated to computing long-range forces in classical molecular dynamics simulations. To this end, we presented a formulation to make MDGRAPE-2 compute the layer potential that is the most time-consuming in a run of the TDBIEM, considering to minimise the amount of data-transfer between MDGRAPE-2 and the host computer. The newly implemented TDBIEM code using three MDGRAPE-2s boards was about 109 times faster than the conventional code in an analysis of 5400 boundary elements and 640 time steps. We also demonstrated that the present hardware acceleration helps to perform large-scale simulations such as light propagation in photonic crystals.

レーザ超音波計測による波形データを用いたクラック決定解析

This work considers the determination of cracks with the time domain elastodynamic boundary integral equation method (BIEM) in laser-ultrasonic non-destructive evaluation problems. We carry out an inverse problem to determine the thermally expanded source from the measured velocimetory waveform data using Green's function of heat conduction problems and that of elastodynamics. With the thermally expanded source thus determined, we can calculate the laser generated elastic wave field numerically using the time domain BIEM. We then represent the unknown crack in the material with shape parameters and compute the normal velocities on the surface of the test piece with BIEM. The shape parameters of the crack are determined by a comparison between the computed waveforms and the corresponding measured data at the measuring points. We show that we can determine the shape and the position of the surface crack in the test piece made of aluminum alloy accurately using time domain BIEM and real data obtained in a laser-ultrasonics measurement.

Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM

This paper presents a transient dynamic crack analysis for a functionally graded material (FGM) by using a hypersingular time-domain boundary integral equation method. The spatial variations of the material parameters of the FGM are described by an exponential law. A numerical solution procedure is developed for solving the hypersingular time-domain traction BIE. To avoid the use of time-dependent Green’s functions which are not available for general FGM, a convolution quadrature formula is adopted for approximating the temporal convolution, while a Galerkin method is applied for the spatial discretization of the hypersingular time-domain traction BIE. Numerical results for the transient dynamic stress intensity factors for a finite crack in an infinite and linear elastic FGM subjected to an impact anti-plane crack-face loading are presented and discussed. The effects of the material gradients of the FGM on the transient dynamic stress intensity factors and their dynamic overshoot over the corresponding static stress intensity factors are analyzed.

The influence of microcrack nucleation on dynamic crack growth––a numerical study

The study deals with the numerical investigation of microcrack nucleation in front of a fast running macrocrack. The computations are based on a time-domain boundary integral equation method. The influence of the microcracks on different parameters such as the stress intensity factor, energy release rate or crack speed are investigated. Results obtained from numerical simulations are discussed with respect to experimental findings.

Time-domain analysis of electromagnetic wave fields by boundary integral equation method

Time-domain analysis of electromagnetic wave fields is popularly performed by the Finite Difference Time-Domain method. Then the Boundary Integral Equation Method (BEM) still has advantage comparing with FDM or FEM type scheme in open boundary problems, moving boundary problems and coupled problems of charge particle and electromagnetic fields. However, the time-domain boundary integral equation method still do not well developed, numerical instability in long time range calculations frequently appear except for special cases. In this paper, a stable scheme of the time-domain boundary integral equation method is presented and numerical example of particle accelerator wake fields is shown.

Stable time-domain boundary integral equation method for axisymmetric coupled charge-electromagnetic field problems

In this paper, a stable time-domain boundary integral equation method for transient electromagnetic wave is presented. Careful numerical treatments of the integral equation for stable solution are especially stated. The formulation is given for axisymmetric electromagnetic field problems. A numerical simulation of particle accelerator wake field is also given as an example.

A new hybrid FDTD-BIE approach to model electromagnetic scattering problems

A new hybrid finite-difference time-domain boundary integral equation (FDTD-BIE) method is proposed. The geometry under consideration is decomposed into several subregions. For some of the bounded subregions the FDTD technique is used to construct an interaction matrix, relating virtual electric and magnetic currents at the boundary of the subregion, for several frequencies at the same time. The remaining bounded and unbounded subregions are described by a boundary integral equation technique. An example is given to validate the new technique.

Two parallel penny-shaped cracks under the action of dynamic impacts

The three-dimensional elastodynamic response of two parallel penny-shaped cracks embedded in an infinite elastic solid under the action of impact loading is investigated. A time-domain boundary integral equation method is used for calculating the time-dependent crack opening displacements and subsequently the dynamic stress intensity factors. Numerical computations are carried out for various geometry parameters. The results are presented in graphical form and discussed. The effect of the locations of the cracks on the dynamic stress intensity factors is presented.

Interaction of two coplanar penny-shaped cracks with transient elastic waves

The three-dimensional elastodynamic response of two coplanar penny-shaped cracks embedded in an infinite elastic solid subjected to transient elastic waves is investigated. A time-domain boundary integral equation method is used for calculating the time dependent crack opening displacements and subsequently the dynamic stress intensity factors. Numerical computations are carried out for various incident transient longitudinal and transverse waves. The results are presented in graphical form and discussed. The effect of the crack distance on the dynamic stress intensity factors is determined.

Analysis of dynamic crack propagation using a time-domain boundary integral equation method

Presented in this paper is a boundary element method which enables the simulation of dynamic crack propagation under arbitrary loading conditions and without any a priori assumptions regarding the crack path. The direction and speed of crack advance are controlled only by a fracture criterion. Fast crack growth is investigated in an unbounded linear elastic body under plane strain or longitudinal shear. For in-plane deformation under mixed mode loading the influence of crack closure on the computed crack path is considered by solving the related contact problem.The starting point is a non-hypersingular time-domain traction boundary integral equation. A collocation method in conjunction with a time-stepping scheme is applied to solve the integral equation numerically. The accuracy of the numerical method is checked by comparison with analytical solutions available for simple situations. Some examples of curved cracks propagating with variable speed in inhomogenous stress fields serve to illustrate the versatility of the method.

Nonlinear Reaction Forces on Oscillating Bodies by a Time-Domain Panel Method

An outline of a time-domain boundary integral equation method (with second-order accuracy) for bodies floating on a fluid is given. It includes the full nonlinear dynamics of a potential flow with a free surface. To validate the method, a partially immersed body is forced to oscillate at a small amplitude for each of three modes. Relevant components of the reaction force on the body are integrated separately, thereby enhancing the accuracy of the hydrodynamic coefficients determined from this force signal. Linear frequency-domain results in a range of frequencies agree well with the present ones for the square, excellently for the circle. Nonlinear fluid-body interaction is simulated for a vertically oscillating square with amplitudes up to 80% of its draft. The force dependence on the motion amplitude provides a confirmation of second-order theory. Furthermore, the triple-frequency force component shows a pronounced cubic dependence on the amplitude.

Axisymmetric dynamic response of a penny-shaped crack to a pair of transient concentrated forces

The three-dimensional axisymmetric elastodynamic response of a penny-shaped crack embedded in an infinite elastic solid subjected to a pair of transient concentrated forces is investigated. The forces are applied on the symmetry axis perpendicular and symmetric to the crack surfaces, including the special case when the forces act precisely on the crack surfaces. A time-domain boundary integral equation method is applied for computing the crack-opening displacement and subsequently the time dependence of the dynamic stress intensity factors. Numerical calculations are carried out for various geometry parameters and the results are discussed. It is found that the location of the applied concentrated forces inducing the highest dynamic stress intensity factors differs from that producing the highest static values.

Transient elastodynamic analysis of a penny-shaped crack

Transient elastodynamic analysis of a stationary penny-shaped crack in an infinite elastic solid is presented. The analysis is based on a non-hypersingular time-domain boundary integral equation method and is fully three-dimensional. An incident, transient and planar displacement pulse of longitudinal or transverse wave-type is taken as applied loading. A collocation method is used for solving the time-domain boundary integral equations. Constant elements are applied away from the crack front, while special elements called “crack front elements” are adopted near the crack front to describe the local behavior of the unknown quantities properly. By using a linear shape function in time and by invoking the causality and the time-translation properties of Green's functions, an explicit time-stepping scheme is obtained for computing the discrete crack opening displacements at the collocation points, from which the transient dynamic stress intensity factors are subsequently extracted. Numerical results are presented for several cases, to analyze the dynamic overshoot phenomena and the effects of the type and direction of incident transient elastic waves on the time history and the spatial variation of the dynamic stress intensity factors.

Pulse shape effects on the dynamic stress intensity factor

The interaction of a transient stress pulse with a penny-shaped crack embedded in an infinite elastic solid is investigated. The front of the incident stress pulse is assumed to be planar and parallel to the crack surfaces. A time-domain boundary integral equation method is applied for computing the time history of the crack opening displacement, from which the time dependence of the dynamic stress intensity factor is subsequently calculated. Numerical calculations are carried out for several stress pulses of different shape and time dependence, to explore the effects of the shape, duration, rise and descent time of a transient stress pulse, or the period and the mean stress of a cyclic stress pulse on the dynamic stress intensity factor. Implications regarding crack surface penetrations or crack surface interactions caused by certain stress pulses are also discussed.

ULTRASONIC SCATTERING ANALYSIS TO CHARACTERIZE NONLINEAR BEHAVIOR OF MATERIAL INTERFACES

Abstract The scattering problem of elastic waves by a three-dimensional inclusion with a nonlinear interface condition is solved. The nonlinear interface condition is represented by using a distribution of nonlinear springs inserted at the matrix-inclusion interface. The springs have perfect elastoplastic characteristics, while the inclusion and the surrounding matrix are linear at all times. Thus, the problem is reduced to the boundary-type dynamic nonlinear problem. A time-domain boundary integral equation method is applied to obtain numerically dynamic behaviors of the nonlinear interface. In numerical examples, a spherical inclusion subjected to an incident plane L-wave is considered. Results as functions of time are obtained for displacements and tractions at the nonlinear interface and the scattered far-field.

Boundary integral equation method for transient analysis of 3-D cavities and inclusions

The time-domain boundary integral equation method is applied to the transient analysis of 3-D cavities and inclusions of arbitrary shape and connectivity. In the numerical implementation, the convolution integrals with respect to time are evaluated analytically and the spatial integrations can be performed independent of the time step number. This implementation leads to an efficient time-stepping algorithm. The methods to evaluate the free term and the singular integrals at an arbitrary edge or corner point are also presented. Some numerical examples dealing with scattering of elastic waves in 3-D infinite and half spaces are presented. The accuracy, versatility and efficiency of the present analysis are established.