Time-domain Boundary Integral Equation Research Articles

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.

Simulation of dynamic crack curving and branching under biaxial loading by a time domain boundary integral equation method

Bifurcation and trifurcation of a fast running crack under various biaxial loading conditions is investigated numerically. The solution procedure for the 2D model in the framework of linear elastodynamics employs a time-domain boundary element method and allows for arbitrary curvilinear crack propagation. Branching events are controlled by the criterion of a critical mode I stress intensity factor while the propagation direction and growth rate of each branch are determined from the criterion of maximum circumferential stress. Numerical results are compared with experimental findings and are discussed with respect to macroscopic and microscopic aspects of dynamic fracture.

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.

Transient elastodynamic antiplane crack analysis of anisotropic solids

Transient elastodynamic analysis of an antiplane crack in anisotropic solids is presented. A time-domain traction boundary integral equation (BIE) method is applied for this purpose. The traction BIE is hypersingular and has the crack-opening-displacement as its fundamental unknown quantity. Unlike the usual time-domain BIE method the present formulation applies a convolution quadrature developed by Lubich (Lubich, C., 1988a,b. Convolution quadrature and discretized operational calculus. Numer. Math. 52, 129–145 (Part I), 413–425 (Part II)) which requires only the Laplace-domain instead of the time-domain Green’s functions. The spatial variation of the crack-opening-displacement is approximated by an infinite series of Chebyshev polynomials which take the local behavior of the crack-opening-displacement at crack-tips into account. By using a Galerkin method, the time-domain BIE is converted into a system of linear algebraic equations which can be solved step by step. Special attention is devoted to the computation of dynamic stress intensity factors of an antiplane crack in generally anisotropic solids. Numerical results for isotropic solids are presented and compared with the well-known analytical results of Thau and Lu (Thau, S.A., Lu, T.H., 1970. Diffraction of transient horizontal shear waves by a finite crack and a finite rigid ribbon. Int. J. Enggn. Sci. 8, 857–874), to check the accuracy and efficiency of the present time-domain BIE method. The effect of the material anisotropy on the dynamic stress intensity factors is analyzed via several numerical examples.

Invariant BEM-SIBC formulations for time- and frequency-domain eddy current problems

We demonstrate that frequency- and time-domain boundary integral equation formulations involving the surface impedance boundary conditions (SIBCs) can be transformed to invariant forms that depend only on the geometry of the problem and do not contain temporary parameters of the field. Thus the integral equations have to be solved only once for a given system of conductors and then the results for any source field (steady state or transient) can be easily obtained. We considered both low- and high-order SIBCs so that the formulations obtained can be applied to a wide range of skin effect problems. A numerical example is included to illustrate the theory.

Rigorous analysis of frequency selective surfaces of finite extent using a hybrid finite difference time domain–boundary integral equation technique

A hybrid finite difference time domain–boundary integral equation approach is used to model frequency selective surfaces of finite extent. Using the finite difference time domain technique, a system matrix is constructed for a box enclosing all relevant features within a unit cell. The open character of the finite frequency selective surface is modeled by coupling a boundary integral equation formulation to the system matrix found with the finite difference time domain approach. A number of relevant examples are studied, and a comparison is made between our new technique and the more classical finite difference time domain technique or the hybrid finite element–boundary integral equation method.

Time-domain BEM for dynamic crack analysis

Time-domain boundary element method (BEM) for elastodynamic crack analysis is presented in this paper. Special attention is devoted to BEM based on traction boundary integral equations (BIEs) in time-domain. Both the hypersingular and the non-hypersingular traction BIEs are dealt with. A time-stepping scheme in conjunction with a collocation method is applied for solving the time-domain BIEs numerically. This scheme uses a linear shape-function in time which allows an analytic time-integration of the system matrix. Two different spatial shape-functions are used in the numerical solution procedure, namely, “crack-tip elements” behind crack-tips while constant elements away from crack-tips. The use of “crack-tip elements” takes the proper local behavior of the crack opening displacements (COD) at crack-tips into account, and it makes an accurate calculation of the dynamic stress intensity factors feasible. The hypersingular integrals arising in the hypersingular time-domain traction BEM can be easily evaluated numerically and analytically by a direct method. Numerical examples for the dynamic stress intensity factors are given to show that the time-domain traction BEM is very accurate and efficient for elastodynamic crack analysis.

On the interaction and branching of fast running cracks—a numerical investigation

Phenomena of dynamic fracture in brittle materials resulting from the interaction of fast running cracks are investigated numerically. These studies concern the influence of mutual shielding on crack paths, the stress intensity factor and crack tip speed variations, the stability of a simultaneous propagation of several cracks, and the interaction of cracks of different size. Furthermore, dynamic crack branching is investigated from a macroscopic point of view. The respective initial boundary value problem of linear elastodynamics is formulated as a system of time-domain boundary integral equations in conjunction with experimentally motivated criteria for crack growth and branching. The numerical evaluation is based on a boundary element method and an appropriate discretization of the fracture and branching criteria. Results obtained from numerical simulations are discussed with regard to the experimental findings.

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.

Time-domain BIE analysis of large three-dimensional electromagnetic scattering problems

A time-domain boundary integral equation (BIE) solution of the magnetic field integral equation (MFIE) for large electromagnetic scattering problems is presented. It employs isoparametric curvilinear quadratic elements to model fields, geometry, and time dependence, eliminating staircasing problems. The approach is implicit, which seems to provide both stability and permits arbitrary local mesh refinement to model geometrically difficult regions without the significant cost penalty explicit methods suffer. Error dependence on discretization is investigated; accurate results are obtained with as few as five nodes per wavelength. The performance both on large scatterers and on low-radar cross section (RCS) scatterers is demonstrated, including the six wavelength NASA almond, two spheres, a thirteen wavelength missile, and a high-Q cavity.

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.

On the large time behaviour of transient dynamic kernels for two‐dimensional wave scattering

AbstractAn essential feature of the transient dynamic kernels for any time domain boundary integral equation is that they decay towards the static kernels for large time increments. The purpose of the paper is to show that the kernels of Mansur and Brebbia (1982), which are expressed in the form of generalized mathematical functions, do exhibit the correct decay property in contradiction to the claims of Israil and Banerjee (1990). Mansur and Brebbia express the fundamental solution wave discontinuity in terms of the generalized functions H(t) and δ(t), the Heaviside and Dirac delta functions, but do not explicitly demonstrate the required long time behaviour. In the 1990 paper the temporal discontinuity of the kernels is expressed by conventional functions in two time regimes and is shown to possess the correct long time behaviour, while it is claimed that the kernels in the 1982 paper do not. It is shown here that, assuming either constant or linear variation in time, usage of the generalized functions in the fundamental solution does yield the static fundamental solution for large time steps, as expected.

Time domain fundamental solution to Biot's complete equations of dynamic poroelasticity. Part I: Two-dimensional solution

This paper develops transient fundamental solution for Biot's full dynamic two-dimensional equations of poroelasticity. An explicit, well-posed Laplace transform domain fundamental solution is obtained for the governing differential equations which are established in terms of solid displacements and fluid pressure. In some limiting cases, the solutions are shown to reduce to those of classical elastodynamics and steady state poroelasticity, thus ensuring the validity of our result. The closed-form transient fundamental solutions both for the limiting case (early time approximation) and for the general case are derived from corresponding ones in the Laplace transform domain by means of Laplace transform techniques. They represent the very first satisfactory fundamental solutions in the real-time domain for full dynamic poroelasticity. Some characteristics of the solutions are investigated. Selected numerical results are presented to demonstrate the features of the waves, and the accuracy of the solution is established by comparing them with Laplace transform domain solutions. To complete the ingredients for developing the BEM, the Betti's reciprocal theorem is extended to the full dynamic poroelasticity, based on which a time domain boundary integral equation is obtained.

2-D scattering by a crack with contact-boundary conditions

The dynamic contact problem of a two-dimensional crack subjected to an incident wave is solved using a time domain boundary integral equation. There are three possible phases in the contact-boundary conditions, namely, separation, slip contact and stick contact. Considered here are three examples: (i) a pre-opened crack subjected to normal incidence of an L wave; (ii) a pre-stressed crack subjected to normal incidence of an L or T wave; (iii) a crack with frictional contact conditions subjected to an obliquely incident T wave. In these calculations, near field solutions are obtained as well as scattered far-fields. It is shown that the Fourier amplitudes of the higher harmonics of the scattered far-fields may be useful to determine the pre-stress, the frictional coefficient and the initial crack-opening displacement.

On fundamental solutions in time-domain boundary element analysis of linear viscoelasticity

This paper presents an efficient method for the computation of the fundamental solutions in time-domain boundary integral equation for viscoelastic problems. The method proposed is based on the Laplace transform and the correspondence principle. The relaxation function is expanded in a sum of exponentials and the transformed fundamental solutions are inverted numerically into real time space. The proposed procedure requires a small computational effort and it is applicable in time-domain boundary element analysis of realistic viscoelastic problems. Numerical results of an example problem show the effectiveness and applicability of the proposed method.

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.