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This work presents a novel scheme to couple the Discrete Element Method (DEM) and the Boundary Element Method (BEM) for the multi-scale modelling in the time domain. The DEM can model discontinuous material at micro scale very well, but it cannot represent infinite domains. Hence, coupling with the BEM is proposed. A formulation employing the DEM and BEM in different subdomains of the same body is presented. There is no overlap between the sub-domains, and the system of equations is derived based on strong equilibrium and compatibility conditions at the interface. The proposed coupling scheme is based on monolithic time integration. The conducted numerical experiments of one-dimensional wave propagation show excellent agreement with the analytical solution. Some spurious wave reflections are observed at the interface, but their effect is quantified and deemed not critical for infinite domains, which are of main interest. Even though the applications for one-dimensional wave propagation are of limited practical engineering interest, this work represents a significant theoretical breakthrough. This paper establishes a reference for future coupling schemes for two- and three-dimensional multi-scale analysis.

In this paper, based on dimension splitting method and the improved interpolating moving least-squares (IMLS) method, a dimension splitting interpolating element-free Galerkin (DSIEFG) method for three-dimensional (3D) transient heat conduction problems is proposed. The main idea of the DSIEFG method is to split a 3D problem domain into a series of related two-dimensional (2D) subdomains. The improved IMLS method is used to construct shape function on 2D subdomains. Finite difference method is used to couple these discretized equations on 2D subdomains and employed in the time domain. Compared with the improved element-free Galerkin (IEFG) method, the advantage of the DSIEFG method is that the essential boundary conditions can be applied directly, which can improve computational accuracy and efficiency. Six examples are chosen to verify the effectiveness and efficiency of the DSIEFG method. The results of DSIEFG are compared with the numerical solutions of the IEFG method, and it is shown that the efficiency and precision of the DSIEFG method are greater than ones of the IEFG method for 3D transient heat conduction problems.

The hydrodynamic analysis for ships is studied based on the three-dimensional (3D) time domain Green function (TDGF) method. The precise integration method (PIM) with variable dimension matrix is developed to evaluate the TDGF and its derivatives. The numerical convergence analysis for TDGF evaluation is carried out. Based on the TDGF method, the radiation problem and diffraction problem are solved by using impulsive response function. The hydrodynamic coefficients, wave exciting forces and motions for Wigley Ⅰ hull form ship are computed to verify the reliability and accuracy of the three-dimensional time domain program. The numerical results obtained by the proposed method show a good agreement with the previous experimental data and other published results.

Phase change problems are investigated numerically in this study. A combined precise time-domain expanding boundary element method (PTEBEM) and the level set method is proposed. The PTEBEM is employed on one hand to solve the temporal-spatial field partial differential equations of the transient heat transfer problem. The radial integral method is used to transform the domain integral of the governing equation into a boundary integral. On the other hand, the level set method which is widely used to track crack trajectories is here employed to handle the complicated topological changing during phase change, to record the position of material phase interface during moving, merging or slipping. The formed numerical framework is further enhanced for higher stability. To demonstrate the accuracy and robustness of the proposed method, a few examples are provided.

In this research, based on three-dimensional (3D) time domain hybrid Green function method, the time-domain nonlinear initial boundary value problem of a ship motion in waves with forward speed is solved. The 3D Rankine source and 3D time domain free surface Green function are adopted to solve flow fields of the inner and the outer domains respectively. Meantime, the instantaneous wet surface during ship motion is divided into boundary elements in every time step, to calculate nonlinear incident wave and static restoring forces. Combining with the linear solution of radiation and diffraction problems, the nonlinear time domain numerical method based on 3D hybrid Green function method for ship motion in waves with forward speed is established. Numerical simulation and analysis are carried out for Wigley and S175 ships with different forward speeds and wave angles respectively by developed numerical solver. The comparison with the experimental results show that the proposed nonlinear numerical method has an obvious improvement in predicting accuracy under complicated conditions such as with higher forward speed and large flare, which proves it has good prospect in engineering applications.

In this paper, we develop a meshless method by using radial basis functions (RBFs) to solve time domain Maxwell’s equations resulting from simulating wave propagation in electromagnetic wave splitter and rotator devices. To simulate wave propagation in these devices, we introduce a perfectly matched layer (PML) to reduce the unbounded physical domain problem to a bounded domain problem. Using PML leads to a multi-physics problem with different governing equations in different subdomains, which makes the simulation quite challenging. By following the idea of leap-frog FDTD scheme, we develop the leap-frog RBF meshless method to solve these complicated coupled modeling equations. Extensive numerical results by using multiquadric RBF and Gaussian RBF are presented to demonstrate the effectiveness of our meshless method.

In this paper both of the elastic-viscoelastic correspondence principle and time stepping method, which can convert viscoelasticity to elasticity, are employed to the boundary element method (BEM) to solve the contact problems between multiple rigid punches and anisotropic viscoelastic foundation. The one based upon the correspondence principle has the restriction on time-invariant boundaries, and hence can only be applied to the cases of complete indentation whose contact region as well as contact status are time-independent. The other based upon the time stepping method is processed step-by-step in time domain, and its boundary conditions may vary in each time step, and hence can be applied to both complete and incomplete indentations. Both methods are valid for the frictionless or frictional contact surface, the punches can be in equilibrium status or in quasi-static sliding condition, and the number of punches and the punch profiles are arbitrary. Benefited by the elastic system used for BEM, both methods hold the feature that no meshes are required on the rigid punches and holes/cracks. Through the iteration procedure and numerical examples presented in this study, we see that the one based upon time-stepping method is much more general and efficient than the other one.

The open multi-processing parallelism was implemented into a serial code to compute a high-order boundary integral equation based on the second Green's identity. on symmetric multiprocessing computer platforms. The parallel code is augmented to a three-dimensional potential numerical wave tank for fully nonlinear wave-body interaction problems in the time domain. The non-uniform rational B-Spline surface was manipulated to define the geometry of the computational boundaries and the distribution of boundary values either as a high-order iso-geometric formulation. The material node approach and fourth-order Runge-Kutta time integration method were compounded for time marching the fully nonlinear free surface boundary. The propagation of a fifth-order Stokes wave was simulated to examine the scalability of the parallel constructs implementation. Afterwards, the second-order Stokes wave interaction with a cylindrical pile was modeled as a practical case study to show the benefits of the parallel code. The results were compared with former experiments and numerical studies.

Oblique water entry of a rigid paraboloid is investigated by fully nonlinear boundary element method. The deadrise angle of the paraboloid increases gradually from zero at the bottom tip and a slenderness coefficient is introduced to symbolize the shape of the paraboloid. Convergence study with respect to time step and element size is carried on to assure the numerical procedure. The free surface elevation and pressure distribution are depicted in time domain. The present solution is compared with the linearized Wagner method and the composite solution with spray jets to investigate the influence of exact fully nonlinear boundary conditions. As there is no sharp edge or tip around the paraboloid, the water flows over the body both from lateral side and down side in oblique entry, the stagnation point relative to the body moves towards the flow-coming direction, and the negative pressure can be observed on both lateral side and down side of the paraboloid. The horizontal speed has great influence on horizontal total force and little influence on vertical total force, while the effect of the slenderness coefficient just has the opposite effect.

This paper presents a comparative study on the motions of a ship advancing in waves using two different three-dimensional time domain boundary element methods: a Transient Free surface Green's Function (TFGF) method and a Rankine Higher Order Boundary Element Method (HOBEM). Three models based on the HOBEM are also considered to identify the dominant factors affecting the accuracy of solutions: (i) the N–K model using Neumann–Kelvin (N–K) linearization, (ii) the N–K+DB model using the N–K linearization for the free surface boundary conditions and the double-body (DB) m-terms in the body boundary condition, (iii) the Rankine HOBEM based on DB linearization. The Wigley I and the Series 60 (CB = =0.7) are taken as study objects. The comparisons show that the Rankine HOBEM based on DB linearization is generally more accurate than the Rankine HOBEM with other two models and the TFGF method. The hydrodynamic coefficients are mainly affected by the m-terms, especially at low frequencies; while the influences of different linearizations of the free surface boundary conditions are negligible. Both the m-terms and the leading terms of steady velocity potential reserved in the free surface boundary conditions are important for motion responses, while the influence of the former is stronger than the latter.

This study considers the stability of time domain BEMs for the wave equation in 2D. We show that the question of stability of time domain BEMs is reduced to a nonlinear eigenvalue problem related to frequency domain integral equations. We propose to solve this non-linear eigenvalue problem numerically with the Sakurai-Sugiura method. After validating this approach numerically in the exterior Dirichlet problem, we proceed to transmission problems in which we find that some time domain counterparts of “resonance-free” integral equations in frequency domain lead to instability. We finally show that the proposed stability analysis helps to reformulate these equations to obtain stable numerical schemes.

An improved meshfree radial point interpolation method (RPIM) is formulated for quasi-static and dynamic analyses of 2D inhomogeneous viscoelastic solids. The presented improved RPIM is formulated in time domain, in which, there is no need for the viscoelastic material to obey the correspondence principle and therefore, all kinds of inhomogeneous viscoelastic materials can be analyzed by the method. The domain integrals, appeared in the proposed RPIM, are computed by the Cartesian transformation method (CTM), which is a truly meshfree integration technique. In the implementation of the CTM no background mesh is required and the domain integrals are computed accurately and efficiently. The accuracy of the method is studied by presenting several numerical examples. The results are compared with available analytic or numerical solutions and very good agreement is observed in all cases.

A radial integration boundary element method is applied to the analysis of two-dimensional scalar wave equation problem. Based on two kinds of fundamental solutions, two types of the integral equation is derived for solution of wave problem. Time domain integrals which are imposed to the solution are implemented using an efficient modified formulation of the radial integration method for concave shapes which was recently introduced in [1] . This boundary only element approach is applied to Newmark and Houbolt time stepping methods. Performance of the implemented formulation is evaluated using several numerical examples.

We discuss the algorithmic aspects of a recently proposed time-domain partition-of-unity boundary element method for wave propagation problems a high frequency on coarse mesh grids. A main challenge is the accurate assembly of the Galerkin matrix for travelling plane-wave basis functions. We discuss the numerical solution of the space-time system and its preconditioning. Numerical results demonstrate the prospects and challenges of the approach.

Numerical stability has been a fundamental challenge in direct time-domain boundary element methods (TD-BEM) for elastodynamics. In this paper, an analytical framework for the evaluation of the critical aspect is presented. By casting a convolution integral-based TD-BEM algorithm in the form of a linear multi-step method with a hybrid amplification matrix and the incorporation of some fundamental characteristics of commonly-used transient Green's functions, a rigorous assessment of the problem is shown to be reducible to a standard spectral analysis in matrix theory by which the stability threshold can be clearly defined. Apt to be relevant to the evaluation of other TD-BEMs, the approach is applied to a regularized time-domain direct boundary element method with optional collocation weights and orders of solution variable projections as illustration. By virtue of the proposed formalism and a systematic parametric study, a proper resolution of the critical aspect for some past schemes as well as the versatility of the generalized TD-BEM algorithm as they pertain to the benchmark finite-domain square-bar and the infinite-domain cavity elastodynamic problems are given as examples.

We propose a time stepping scheme for the space-time systems obtained from Galerkin time-domain boundary element methods for the wave equation. Based on extrapolation, the method proves stable, becomes exact for increasing degrees of freedom and can be used either as a preconditioner, or as an efficient standalone solver for scattering problems with smooth solutions. It also significantly reduces the number of GMRES iterations for screen problems, with less regularity, and we explore its limitations for enriched methods based on non-polynomial approximation spaces., 15 pages, 16 figures

To advance Time-Domain Boundary Element Methods (TD-BEMs), a generalized direct time-integration solution method for three-dimensional elastodynamics is presented in this paper. On the basis of a general decomposition of time-dependent point-load Green's functions into a singular and regular part, a regularized boundary integral equation for the time domain is formulated and implemented via a variable-weight multi-step collocation scheme that allows for different orders of time projection for the boundary displacements and tractions. The benefits and possibilities of improved performance by suitable collocation weights and the solution projection choices are illustrated via two benchmark finite-domain and infinite-domain problems.

A fully nonlinear Numerical Towing Tank (NTT) is developed to simulate ship waves based on non-uniform rational B-spline (NURBS) surface and potential theory in the time domain. Fully nonlinear free surface boundary conditions and the exact body boundary condition are applied to solve the boundary value problem. The free surface and the ship hull surfaces are described precisely using NURBS formulation. Mixed Eulerian–Lagrangian (MEL) method is used to update the exact free surface based on material node approach. Isoparametric direct boundary integral equation (BIE) is used to compute the flow field. The intersection of the free surface and the rigid walls are found using an adaptive methodology based on NURBS and double nodes approach at each time step. To retain the open sea condition, a sponge layer is adopted on the exact free surface adjacent the tank's walls. To examine the accuracy of NURBS surface, the NURBS description of the ship hull for different orders of the basis function and different numbers of nodal points are compared with the exact surface. To verify the present computational algorithm, the solutions are compared with the experimental records and the prior numerical study for a Wigley hull. Propagation of the ship wave within the computational domain is studied due to ship advance in different water depths.

An efficient high order plane wave time domain algorithm is presented for analyzing the transient scattering from three dimensional electrically large conducting objects. This method uses a set of hierarchical divergence-conforming vector basis functions to accurately represent the current distribution on the perfect electrically conducting (PEC) surface. The higher order functions can significantly reduce the number of unknowns without compromise on the accuracy. The time domain combined field integral equation (TD-CFIE) is then discretized using the hierarchical divergence-conforming vector basis functions and shifted Lagrange polynomial functions in spatial and time domain, respectively. The final matrix equation can be accelerated using the plane wave time domain (PWTD) algorithm. Finally, a parallel algorithm that can execute on a distributed-memory parallel cluster is developed, which provides an appealing avenue for analyzing the transient scattering from three-dimensional electrically large complex PEC objects. Numerical examples are given to demonstrate the accuracy and efficiency of the method.

Due to the “wave velocity error” between the discretized model and continuum systems, the accuracy of numerical results using standard finite element method (FEM) in time domain acoustic problems is unsatisfactory with increasing frequency. Such wave velocity error is strongly related to the balance between the “stiffness” and “mass” of discretized systems. By adjusting the location of integration point of mass matrix, the redistribution of the mass is able to “tune” the balance between the stiffness and mass. Thus, the wave velocity error can be minimized in time domain acoustic problems with a balance system. On the other hand, it is found that the stability of discretized model of time domain acoustic problems can be improved by the softened stiffness. Furthermore, it is found that the balance between the smoothed stiffness and mass can also be achieved with the tuning of integration point r in the mass matrix.

The paper deals with the direct time domain calculation of a transient electric field generated by a ground penetrating radar (GPR) dipole antenna and transmitted into the lossy half-space. The space–time dependent current along the dipole is governed by the Hallen integral equation which is numerically solved by means of the Galerkin–Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM). Provided that the current distribution along the GPR antenna is determined, one may compute the related electromagnetic field transmitted into the lossy ground by solving the field integral formulas. Some illustrative computational examples related to the transmitted field into the lower half-space are reported in this paper.

A time domain boundary element method (TBEM) is applied to predict the horn effect. As the time response calculated by the time domain boundary integral equation contains the resonance components, when transformed to the frequency domain, the result will corrupt at the characteristic frequencies. To overcome this problem, a Burton–Miller-type combined time domain integral equation in half-space is applied. The resonance components are excluded in the time domain calculation, thus the corruptions are avoided in the frequency domain. As a result, the horn effect can be predicted very well at all frequencies. Compared to the frequency domain boundary element method for predicting the horn effect, the TBEM is more efficient due to the lower cost of forming coefficient matrices and solving equations. A numerical simulation is carried out to demonstrate the efficiency of the TBEM, and two experiments are conducted to validate the proposed method in predicting the horn effect. Both numerical and experimental results indicate that the proposed method is reliable and efficient in predicting the horn effect.

To analyze the transient electromagnetic (EM) scattering from the composite structure containing inhomogeneous dielectric volumes and perfect electrically conducting (PEC) bodies efficiently and flexibly, a marching-on-in-time (MOT) based time domain non-conformal volume surface integral equations (TD-VSIE) method is proposed. The non-conformal volume integral equation scheme and surface integral equation discontinuous Galerkin (IEDG) method are combined to realize the non-conformal meshes discretization for composite objects. For some composite objects containing fine structure or inhomogeneous dielectric materials, notable reduction in the number of the unknowns can be obtained without sacrificing accuracy. Moreover, a hybrid mesh scheme is introduced to further improve the efficiency. Several numerical results are proposed to demonstrate the validity and flexibility of the proposed scheme.

Scaled Boundary Finite Element Method (SBFEM) and a temporally adaptive algorithm are combined to solve viscoelastic problems. By expanding variables at a discretized time interval, a spatially and temporally coupled viscoelastic problem is decoupled into a series of recursive spatial problems, which are solved by SBFEM, the computing accuracy in the time domain is controlled via a self-adaptive process. For the cyclic symmetric structures, the cyclic symmetry is exploited to reduce the computational expense of SBFEM, both the eigenvalue and system equations of SBFEM are partitioned into a number of smaller independent problems, which are solved by a partitioning algorithm. Two numerical examples are given to verify and illustrate the proposed approach.

A deterministic-stochastic transient study of Ground Penetrating Radar (GPR) dipole antenna radiating in a presence of a two-media configuration is carried out in the paper. A deterministic direct time domain formulation is based on the corresponding space-time Hallen integral equation. The numerical solution is carried out via the improved space-time variant of the Galerkin-Bubnov Indirect Boundary Element Method (GB-IBEM). The Stochastic-Collocation (SC) method is then applied to determine accurate confidence intervals due to the random variations of GPR input parameters. Once obtaining the current along the dipole antenna, it is possible to calculate other parameters of interest for GPR dipole antenna behavior, such as the field reflected from the interface of two media, or the field transmitted into a lower half-space. Some illustrative numerical results for the transient current along the dipole antenna and transient electric field transmitted into the lower half-space are given.

Based on the Biot׳s dynamic equations, the BEM in time-domain for the dynamic analysis of the poroelastic material subjected to a force source is described in this paper. Some methodologies treated to integral of the Somigliana׳s representative are revealed by authors, using the Green׳s functions in U-P formulation in time-domain, which fast and slow compressional waves has been decoupled. The treatment procedure which the integral variables regarding the fluid phase are transformed appropriately is described. The discretization treatment of the boundary integral equations is expounded in detail. The relevant numerical examples for BEM of the poroelastic material are performed by authors, the results are shown in the form of charts. The reasonable comparisons of the current and the previous solutions are also made in this paper. The treatment approach of stability and convergence as well as their dependence on time step are explained also. The pertinent descriptions may have certain effect to analysis on dynamic issues of other source, such as dislocation, dipole for the poroelastic material. Since the numerical investigation of porodynamic problems in time-domain has been rarely, further researches on the dynamic numerical analysis for the poroelastic material may benefit from the approach of this paper in future.

Based on a FE–BE coupling algorithm, an automatic implementation procedure for the coupling of the ABAQUS with a self-written linear elastic BE code is introduced for dynamic elastoplastic problems. User subroutine (UEL) is developed to enable the incorporation of the BE capabilities within the ABAQUS. Each closed BE domain in the mixed FEM/BEM model is defined as a finite super-element for which the effective stiffness and effective forces are generated by the BE code and hence can be assembled to the global FE formulation. The user can not only benefit from the powerful pre- and post-disposal functions of the ABAQUS, but also deal with systems with infinite extension by using the BEM as a supplement. Basic steps of the automatic coupling procedure are explained and necessary background information is provided. The analysis is conducted through several examples regarding 2D time domain responses. The results of the analysis document a very good agreement between the present solutions and analytical and other numerical results, confirming thus a successful implementation of the developed automatic coupling procedure.

A weighting-iteration method in the time domain is developed to calculate the scattered waves from a complex-shaped scatterer. The incident waves can be mono-frequency or multi-frequency, and the complex object includes sharp edges and dramatic variations in geometry. The solid angles on the boundary elements of a complex-shaped scatterer are generally reduced to below the standard value of 0.5 for points on a smooth part of the boundary. These reduced solid angles destroy the convergence history during the iteration process in the time domain. A weighting function associated with the variation of solid angles is introduced to robust and rapid convergence in the time domain. The new method is used to calculate the scattering from a cube with sharp edges and an indented surface. The weighting function speeds up the convergence history to reach a robust convergence for both mono- and multiple-frequency incident waves.

In this work, the application of a meshfree method based on the generalized finite differences (GFD) method is developed for two dimensional analysis of coupled non-Fickian diffusion–elasticity. The two dimensional analyzed domain is subjected to shock loading in the problem. The equations of motion are transferred to Laplace domain by Laplace-transform technique and descritized using the presented meshfree method. The obtained results in Laplace domain are transferred to time domain using Talbot Laplace inversion technique for studying on the dynamic behaviors of displacements and molar concentration. It is found that the molar concentration diffuses through 2D domain with a finite speed similar to elastic wave. The propagation of mass diffusion and elastic waves are obtained and discussed at various time intervals. The distribution of molar concentration and displacements along “ x ” and “ y ” directions are illustrated at various time intervals for certain points on both axes.

Based on the two-dimensional Lagrange series interpolation, the formulation of the Finite Block Petrov–Galerkin (FBPG) in the weak form is presented in this paper. In this case, the first order of partial differentials are only needed in the weak form governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate ( x o y ) to the normalized coordinate ( ξ o η ) with 8 seeds. Time dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin׳s inversion method is used to determine all the physical values in the time domain. Illustrative numerical examples are given and comparisons have been made with either analytical solutions or other numerical solutions including meshless method and the Finite Element Method (ABAQUS).