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.

Analysis of dynamic crack propagation in two-dimensional elastic bodies by coupling the boundary element method and the bond-based peridynamics

A novel method for predicting dynamic crack propagation based on coupling the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed in this work. The special feature of this method is that it can take full advantages of both the BEM and PD to achieve a higher level of accuracy and efficiency. Based on the scale of the structure and the location of cracks, the considered domain can be divided into a non-cracked region and a cracked region. For the non-cracked region, a meshfree boundary-domain integral equation method (meshfree-BDIEM) is employed in the analysis to reduce the dimension by one and to increase the computational efficiency. The bond-based PD is applied to simulate the cracked region, which can model the initiation and propagation of the cracks automatically. The boundary nodes from the BEM on the interfaces can interact with the material points from the PD directly. By using the displacement continuity and force equilibrium conditions on the interfaces, a combined model is obtained by merging the mass, stiffness, and force matrices from each region of the domain. Both the implicit and explicit BEM-PD coupled solution methods can deliver accurate results without inducing the ghost forces. Several benchmark problems of dynamic crack propagation have been modeled by using the method, which demonstrate that the developed BEM-PD coupled approach can be an efficient numerical tool to model the dynamic crack propagation problems.

In-Plane Free Vibration of Circular and Annular FG Disks

The analysis of the in-plane free vibration of the circular and annular FG disks by a meshfree boundary-domain integral equation method is presented in this paper. The material properties of the disks are assumed to vary in the radial direction obeying an exponential law. Based on the two-dimensional linear elastic theory, the motion equations of the FG disks are derived by using the static fundamental solutions. Radial integration method as an efficient tool is adopted to treat the domain integrals which are raised due to the material inhomogeneous and inertial effects. The natural frequencies and associate mode shapes are calculated for the FG disks with combinations of free and clamped boundary conditions. Parametric studies are also conducted to study the effects of the material gradients, radius ratios and boundary conditions on the frequency of the FG disks.

Dynamic behaviors of tapered bi-directional functionally graded beams with various boundary conditions under action of a moving harmonic load

In this paper, the dynamic response of beams made of bi-directional functionally graded materials (FGMs) with varied cross-sections subjected to a moving harmonic load has been studied by a developed boundary-domain integral equation method based on the two-dimensional linear elastic theory. It is assumed that the elastic modulus and mass density of the FG beams vary separately following each exponential distribution parameters through longitudinal or transvers direction. Then, the elastostatic homogeneous fundamental solution was substituted into the dynamic governing equation for deriving the vibration boundary-domain integral equation of the tapered FG beams. A meshfree scheme can be achieved by applying the radial integration technique to transform the domain integrals into boundary integrals. Houbolt's algorithm is employed to obtain the dynamic response of the tapered FG beams. The effectiveness of the present method applied to the non-uniformed FG beams is verified by comparing numerical results with those available for free vibration of tapered beams of linearly variable width or depth. A detailed parametric study is carried out to investigate the influences of the taper ratios, material gradients, gradation directions and boundary conditions as well as the moving load frequencies and velocities on the dynamic behaviors of tapered FG beams.

A meshfree boundary-domain integral equation method for free vibration analysis of the functionally graded beams with open edged cracks

A dynamic analysis may be required either because a crack is excited by time dependent loads or because a crack under static loading conditions propagates so rapidly that the effects of the inertia forces are important and the inertia effects cannot be neglected. In this paper, free vibration of the functionally graded beams with open edged cracks are analyzed by a meshfree boundary-domain integral equation method. Elastostatic fundamental solutions are used as weight functions to generate the weighted residual statements of the equations of motion. Fundamental solutions are obtained by considering the inhomogeneous and inertia effects. Numerical results compared well with that of the analytical methods. Comprehensive parametric study investigates the effects of the material gradients and directions, crack length and depth ratios, as well as boundary conditions on the free vibration responses on the cracked FG beams, which demonstrate the present method states high efficiency and accuracy. Besides, the developed method can be used to identify the crack size and location of the FG beams.

Forced vibration analysis of functionally graded beams by the meshfree boundary-domain integral equation method

Forced vibration of two-dimensional functionally graded beams is studied in this paper by a developed meshfree boundary-domain integral equation method. Material properties of the functionally graded beams are assumed varying continuously either in longitudinal or transvers direction following the exponential function. The boundary-domain integral equations are derived by using the elastostatic fundamental solutions based on the two-dimensional elastic theory. Radial integral method (RIM) is employed to transform the domain integrals into boundary integrals. A meshfree scheme is achieved through assuming the displacements and accelerations in the domain integrals by a combination of the radial basis function and polynomials with time dependent coefficients. Wilson-θ, Houbolt as well as two kinds of damped Newmark's algorithms are applied to accomplish the time integration. The forced vibration of the functionally graded beam subjected by the harmonic loading and transient loading are investigated in detail. Numerical examples demonstrate that the four mentioned time integral schemes are all adapted well to the developed meshfree boundary-domain integral equation method for analyzing the forced vibration of homogeneous structures. For the analysis of FG structures, it is shown that the damped Newmark's algorithm can achieve more stable and accurate results.

An integral equation method to recover non-additive and non-separable heat source without initial temperature

When the unknown heat source is a non-additive and non-separable function of space–time it is very difficult to recover it. The paper proposes a more difficult mixed type inverse heat conduction problem by simultaneously recovering heat source term and unknown initial temperature, under extra measured data of boundary heat fluxes and a final time condition. We derive a domain/boundary integral equation method (DBIEM), and two different types adjoint Trefftz test functions are used to develop two different numerical algorithms, namely the DBIEM1 and the DBIEM2, which are effective and robust to recover the unknown heat source and initial value. In the adjoint Trefftz test functions we introduce a scaling factor and the multiple-scales are introduced in the trial functions, which play key roles to stabilize the resulting linear system. When we add a large absolute and relative noise up to 10% on the supplementary data, the present numerical methods are quite stable against large noise, and with a fast convergent solution of the linear system to determine the expansion coefficients, typically from five to twenty steps. It is remarkable that we can recover the general heat source function and unknown initial temperature without needing of internal measurement of temperature.

A time step amplification method in boundary face method for transient heat conduction

A time domain boundary integral equation method, which is named as quasi-initial condition method, is applied in this paper to solve the transient heat conduction problem. In conventional implementations, however, this method suffers from a numerically unstable problem when the time step is small. To improve the numerical stability of the method, a time step amplification method is proposed. In the proposed method, an amplified time step is adopted to compute the temperature and the flux at the virtual time point. The boundary condition at that virtual time point is determined through a linear interpolation by the conditions at the current time step point and the quasi-initial time. Furthermore, the heat generation in the virtual time step is assumed to be constant which is the same as that in the real time step. The temperature and the flux at the current step time point are then computed through a linear interpolation over the time interval. A short but not rigorous deduction of this method is presented to show that this method is valid in solution to problems in which the temperature and the flux vary linearly respect to time. Numerical examples further demonstrate the numerical stability of the proposed method.

Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method

Free vibration of the functionally graded (FG) sandwich beams are studied by a meshfree boundary-domain integral equation method. Two sandwich beams, namely, FG core with homogeneous face sheets and homogeneous core with FG face sheets are considered in the study. Based on the two-dimensional elasticity theory, the boundary-domain integral equations are derived by applying the elastostatic fundamental solutions. Radial integration method is employed to transform the domain integrals related to material nonhomogeneity and inertia effect into boundary integrals, hence a meshfree scheme involving boundary integrals only is achieved. Each layer in the sandwich beam is modeled separately and the equilibrium and compatibility conditions are enforced at the interfaces to derive the equations for the whole sandwich beam. Extensive parametric study is presented to examine the influence of material composition, material gradient, layer thickness proportion, thickness to length ratio and boundary conditions on the free vibration of FG sandwich beams.

A boundary-domain integral equation method for solving convective heat transfer problems

In this paper, a new boundary-domain integral equation for solving energy equation in convective heat transfer problems is derived. The derived integral equation accounts for fluid velocity which is computed using the velocity boundary-domain integral equations presented in [2,3] (X.W. Gao, 2004, 2005). Numerical implementation of these integral equations is discussed for steady incompressible fluid flows, in which the velocity and pressure integral equations are uncoupled from the temperature equation, and thus can be solved separately. By substituting the velocity, velocity gradient, and pressure integral equations into the energy integral equation, temperatures can be computed without iterative processes. Two numerical examples for 2D problems are given to validate the proposed method.

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.

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.

Three-dimensional dynamic stress analysis on a penny-shaped crack interacting with a suddenly transformed spherical inclusion

In this paper, the interaction between a penny-shaped crack and a near-by suddenly transformed spherical inclusion in 3-dimensional solid is investigated to assess the dynamic effect of the transformation. To simplify the solution procedure, the current problem is divided into two sub-problems by using the superposition principle. A time domain boundary integral equation method (BIEM) is adopted to evaluate the stress and displacement fields. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. The impact effect of the spherical inclusion when it experiences a pure dilatational eigenstrain on the penny-shaped crack is studied. The relationship between the relative location of the inclusion and its impact effect on the time history of the Mode I crack intensity factor is discussed in detail.

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.

Computation of dynamic stress intensity factors by the time domain boundary integral equation method-I. Analysis

Application of the direct time domain boundary integral equation method (BIEM) to the solution of a number of elastodynamic crack problems is presented. In this part I, we describe the analysis which includes the basic governing differential equations, their transformation to boundary integral equations and the numerical solution procedure of the discretized form of the BIE. In addition to the usual constant interpolation in space and time of tractions and displacements on the boundary, a new boundary element which incorporates quadratic variation in space and linear variation in time (of the traction and displacements) is developed. These two boundary elements are implemented in a computer code and are used in the examples described in part II. A consistent method of estimation of the dynamic stress intensity factors in conjunction with the two types of the elements mentioned is described. No special singular crack tip elements are used other than the quadratic isoparametric quarter-point boundary elements for modeling the crack tip singularities. Subsequent results show that the numerical solution is accurate and has good convergence properties with regard to mesh refinement and time step size.

Computation of dynamic stress intensity factors by the time domain boundary integral equation method—II. Examples

Application of the direct time domain boundary integral equation method (BIEM) to the solution of a number of elastodynamic crack problems is presented. The analytical and numerical formulation has been detailed in part I. In this part II we give the details of some examples solved using the time domain BIEM. The examples considered include those involving semi-infinite cracks and finite length cracks in infinite bodies and finite bodies with finite cracks as well as time dependent loading. The numerical results obtained are compared with available analytical and numerical results. It is found that both of the types of elements discussed in part I model the crack tip displacement fields well when no reflected waves are involved. The QL element is slightly more accurate but requires significantly more computer CPU time and memory. An example involving a discontinuously loaded semi-infinite crack displays the wave propagation features quite well using both types of elements. The effects of finite cracks and wave interaction with the finite boundaries of the specimen are also found to be modeled quite well using the CC element but some oscillation in the computed SIF values at later times is observed when these problems are solved using the QL elements. In the case of a finite crack in an infinite body the discontinuity in the slope of the SIF history curve predicted by Thau and Lu is successfully reproduced by the present BIEM whereas it appears that most other numerical results published so far do not seem to be able to model this feature well. For each case some typical boundary meshes used and the stress intensity factor history is presented using the two types of elements and different time step sizes where appropriate. Values of the computer memory and the execution time required are presented in the form of graphs. It is concluded that the time domain BIEM is a viable technique for solving dynamic crack problems using the CC elements but requires further investigation of the numerical oscillations at later times when the QL elements are used.