Time Domain Boundary Element Formulation Research Articles

A new time-domain boundary element formulation for generalized models of viscoelasticity

The contribution is concerned with the novel algorithmic formulation for generalized models of viscoelasticity under quasi-static conditions within the framework of the boundary element method (BEM). The proposed update algorithm is constructed for a generic rheological model of linear viscoelasticity that can either be straightforwardly simplified to recover the basic Kelvin and Maxwell models or readily furthered towards the generalized models of viscoelasticity through the serial or parallel extensions. In contrast to the scarce existing rate formulations of inelasticity developed within BEM, the put forward non-iterative formulation exploits the hybrid semi-implicit update of strain-like kinematic history variables. The challenge arising from the indispensable domain integrals is overcome through the mesh-free Cartesian transformation method (CTM), complemented by the radial point integration (RPIM) technique. The excellent performance of the proposed approach is demonstrated in comparison with the corresponding analytical and finite element results for boundary-value problems with uniform and non-uniform strain fields under different representative modes and types of loading.

Development of TD-BEM Formulation for Dynamic Analysis for Twin-Parallel Circular Tunnels in an Elastic Semi-Innite Medium

In order to simulate the propagation process of subway vibration of parallel tunnels in semi-infinite rocks or soils, time domain boundary element method (TD-BEM) formulation for analyzing the dynamic response of twin-parallel circular tunnels in an elastic semi-infinite medium is developed in this paper. The time domain boundary integral equations of displacement and stress for the elastodynamic problem are presented based on Betti’s reciprocal work theorem, ignoring contributions from initial conditions and body forces. In the process of establishing time domain boundary integral equations, some virtual boundaries are constructed between finite boundaries and the free boundary to form a boundary to refer to the time domain boundary integral equations for a single circular tunnel under dynamic loads. The numerical treatment and solving process of time domain boundary integral equations are given in detail, including temporal discretization, spatial discretization and the assembly of the influencing coefficients. In the process of the numerical implementation, infinite boundary elements are incorporated in time domain boundary element method formulation to satisfy stress free conditions on the ground surface, in addition, to reduce the discretization of the boundary of the ground surface. The applicability and efficiency of the presented time domain boundary element formulation are verified by a deliberately designed example.

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

Generalised Convolution Quadrature with Runge‐Kutta methods for Acoustic Boundary Elements

AbstractIn time domain boundary element formulations the convolution in time can be handled by analytical integration or the convolution quadrature method. Here, the generalised convolution quadrature method is applied based on Runge‐Kutta methods. This method allows a variable time step size and the usage of higher order methods in time. The example shows that the variable time step size preserves the convergence order for non‐smooth solutions. The order of convergence is restricted by the low order spatial discretisation.

Dynamics-based analytical solutions to singular integrals for elastodynamics by time domain boundary element method

The singularities in 2-D time domain boundary element (TD-BEM) formulation for elastodynamics are divided into three categories: the wave front singularity, the space singularity and the dual singularity. A fully analytical procedure for dealing with the three singularities is proposed by adopting the concept of the finite part of an integral (Hadamard principle integral). In order to reduce the computation time, the conventional numerical procedure is adopted for the non-singular integrals in 2-D TD-BEM formulation. Therefore, the algorithm including the fully analytical procedure for dealing with singular integrals and the numerical procedure for dealing with non-singular integrals is implemented in this study. Two examples, 1-D rod and 2-D cavity representing the problems for the finite domain and the infinite domain respectively, are chosen to verify the effectiveness of the proposed algorithm. It shows that the results obtained from the proposed algorithm agree with the analytical solutions with good accuracy, indicating that the proposed algorithm is applicable for elastodynamics in both finite and infinite domains.

A coupled finite element/equivalent source formulation for transient structural-acoustic problems

Recent research suggests that instability problems that occur in the time domain boundary element formulations are manifestations of nonuniqueness difficulties, and that remedies similar to those used in frequency domain formulations can address the difficulties and improve solution stability. For frequency domain equivalent source formulations, the difficulties can be addressed by combining simple and dipole sources together to form “tripole sources”. The basic goals of the current research are to develop an analogous equivalent source formulation for transient acoustic boundary value problems and to combine it with structural finite element analyses to solve transient coupled structural-acoustic problems. A brief description of the equivalent source formulation is given along with several validation cases for acoustic boundary value problems, including one with a closed boundary surface where nonexistence difficulties should occur. The formulation requires convolution summations, which are independent of each other and a brief discussion is given of how the computations can be parallelized. A brief description is also given of the coupled finite element/equivalent source formulation for transient structural-acoustic boundary value problems. Several examples are given to validate the formulation and to demonstrate that the tripole source formulation for the equivalent sources is more stable than the simple source version.

Time domain boundary element formulation for partially saturated poroelasticity

Abstract Based on the Mixture theory and the principles of continuum mechanics, a dynamic three-phase model for partially saturated poroelasticity is established as well as the corresponding governing equations in Laplace domain. The three-dimensional fundamental solutions are deduced following Hormander's method. Based on the weighted residual method, the boundary integral equations are established. The boundary element formulation in time domain for partially saturated media is obtained after regularization by partial integration, spatial discretization, and the time discretization with the Convolution Quadrature Method. The proposed formulation is validated with the semi-analytical one-dimensional solution of a column. Studies with respect to the spatial and temporal discretization show its sensitivity on a fine enough mesh. A half-space example allows to study the wave fronts. Finally, the proposed formulation is used to compute the vibration isolation of an open trench.

Runge–Kutta convolution quadrature for the Boundary Element Method

Time domain boundary element formulations can be established either directly in time domain or via Laplace or Fourier domain. Somewhere in between are the convolution quadrature based boundary element formulations which utilize the Laplace domain fundamental solution but establish a time stepping procedure. Up to now in applications mostly backward differential formulas of second order are used as the underlying multistep method. However, in recent mathematical literature also Runge–Kutta methods have been applied. Here, the use of Runge–Kutta methods is explained in detail and some numerical studies are given. In these studies the backward difference based procedures are compared to Runge–Kutta methods for a non-smooth problem. An ℓ2 norm of the error is used as the basis of comparison, the convergence of which is investigated theoretically as well. The results confirm that the usage of the new techniques is preferable with regard to less numerical oscillations in the solution and better representation of wave fronts.

An accelerated symmetric time-domain boundary element formulation for elasticity

Wave propagation phenomena occur often in semi-infinite regions. It is well known that such problems can be handled well with the boundary element method (BEM). However, it is also known that the BEM, with its dense matrices, becomes prohibitive with respect to storage and computing time. Focusing on wave propagation problems, where a formulation in time domain is preferable, the mentioned limit of the method becomes evident. Several approaches, amongst them the adaptive cross approximation (ACA), have been developed in order to overcome these drawbacks mainly for elliptic problems. The present work focuses on time dependent elastic problems, which are indeed not elliptic. The application of the presented fast boundary element formulation on such problems is enabled by introducing the well known Convolution Quadrature Method (CQM) as time stepping scheme. Thus, the solution of the time dependent problem ends up in the solution of a system of decoupled Laplace domain problems. This detour is worth since the resulting problems are again elliptic and, therefore, the ACA can be used in its standard fashion. The main advantage of this approach of accelerating a time dependent BEM is that it can be easily applied to other fundamental solutions as, e.g., visco- or poroelasticity.

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

AbstractThe boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known.To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels.Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.

New Fast Convolution Algorithm in Boundary-Element Methods for Two- and Three-Dimensional Linear Soil Consolidation Analysis

Over the last 25 years , the time domain boundary element formulations for the linear consolidation theory of Biot involving fully coupled governing differential equations of flow through porous media and those of elastic deformation of porous skeleton have been fully developed and implemented for both single-region and multiregion two-dimensional plane strain, axisymmetric and three-dimensional problems. However, this storage-based convolution method used in those developments was not found to be suitable for solving large practical problems because of the substantial computer disk space requirements. In order to find a better alternative, an accurate integration-based scheme was developed by the present writers and co-workers, in which, the storage was eliminated by accurately recalculating the summation (involved in the time convolution) of the right-hand side at each step during the time marching process. Although this work was not published in any literature, by using this type of approach, solving l...

A time-domain boundary element formulation for the dynamic analysis of non-linear porous media

This paper presents an original time-domain boundary element formulation for the dynamic analysis of porous media. Integral equations for displacements, stresses and pore-pressures, based on non-transient fundamental solutions are considered. Elastoplastic models are also dealt with by the present methodology, extending the applicability of boundary elements to model complex porodynamic problems. At the end of the paper, a discussion concerning two numerical examples is presented, illustrating the potentialities of the new procedure.

Wave propagation in a simplified modelled poroelastic continuum: fundamental solutions and a time domain boundary element formulation

AbstractIn finite element formulations for poroelastic continua a representation of Biot's theory using the unknowns solid displacement and pore pressure is preferred. Such a formulation is possible either for quasi‐static problems or for dynamic problems if the inertia effects of the fluid are neglected. Contrary to these formulations a boundary element method (BEM) for the general case of Biot's theory in time domain has been published (Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. Lecture Notes in Applied Mechanics. Springer: Berlin, Heidelberg, New York, 2001.). If the advantages of both methods are required it is common practice to couple both methods. However, for such a coupled FE/BE procedure a BEM for the simplified dynamic Biot theory as used in FEM must be developed.Therefore, here, the fundamental solutions as well as a BE time stepping procedure is presented for the simplified dynamic theory where the inertia effects of the fluid are neglected. Further, a semi‐analytical one‐dimensional solution is presented to check the proposed BE formulation. Finally, wave propagation problems are studied using either the complete Biot theory as well as the simplified theory. These examples show that no significant differences occur for the selected material. Copyright © 2005 John Wiley & Sons, Ltd.

Time domain surface impedance concept for low frequency electromagnetic problems—Part II: Application to transient skin and proximity effect problems in cylindrical conductors

A time domain boundary element formulation employing the surface impedance boundary conditions (SIBCs) is developed for the 3-dimensional transient eddy current problem of cylindrical conductors. SIBCs of different orders of approximation are implemented using the perturbation technique in the small parameter proportional to the ratio of the skin depth and characteristic size of the conductor cross-section. The formulation consists of a set of time domain surface integral equations that have identical left-hand sides and can be solved using the same program procedure. The number of equations is determined by the order of approximation of the SIBC, namely: solutions in the perfect electrical conductor (PEC) limit (lowest order) and in the so-called Rytov approximation (highest order) are given by one and four equations, respectively. It is demonstrated that each equation admits separation of variables into space and time components, a property that significantly reduces computational costs compared with traditional time domain formulations that require the integral equations to be solved at each time step. For the purpose of validation, a test problem is solved by the proposed formulation and by the ‘original’ BEM based on the time-dependent fundamental solution. Conditions of applicability are discussed and the effect of such factors as the shape of the incident current pulse and proximity effect are considered.

An efficient direct BEM numerical scheme for phase change problems using Fourier series

Due to the moving boundary nature of the solidification and melting problems, the application of the standard time domain boundary element formulation of the problem presents the difficulty of the numerical evaluation of the history dependence, representing more and more work as time proceeds. In that regard, we present an algorithm based on a Green's function which allows a Fourier representation of the domain integral in the reinitialization scheme of the direct boundary element method formulation of the problem, from which it is possible to carry out the history integration in a recursive manner. To obtain the reinitialization approach, in the integral representation formula, the Green's function corresponding to zero temperature in a box containing the original domain will be used, instead of using the classical free space fundamental solution. This Green's function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain in consideration. It can therefore be used in the integral representation formula of the heat equation (direct formulation) to obtain the solution of a heat problem in such domain. The mentioned Green's function can be obtained by the images method, and the resulting source series can also be rewritten in terms of a double Fourier series, that will be used in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals. In this work a new reinitialization scheme has been applied to solve moving boundary problems, to detect the location of the phase-change interface in solidification and melting problems. The numerical results were compared with analytical solutions (in the case, where they were available) showing a good agreement between the results, giving special emphasis to the interface location and the temperature distribution.

Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids

The dynamic responses of fluid-saturated semi-infinite porous continua to transient excitations such as seismic waves or ground vibrations are important in the design of soil-structure systems. Biot's theory of porous media governs the wave propagation in a porous elastic solid infiltrated with fluid. The significant difference to an elastic solid is the appearance of the so-called slow compressional wave. The most powerful methodology to tackle wave propagation in a semi-infinite homogeneous poroelastic domain is the boundary element method (BEM). To model the dynamic behavior of a poroelastic material in the time domain, the time domain fundamental solution is needed. Such solution however does not exist in closed form. The recently developed ‘convolution quadrature method’, proposed by Lubich, utilizes the existing Laplace transformed fundamental solution and makes it possible to work in the time domain. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time-stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method. Finally, two examples show both the accuracy of the proposed time-stepping procedure and the appearance of the slow compressional wave, additionally to the other waves known from elastodynamics.

A new time domain boundary element formulation for generalized dynamic coupled thermoelasticity

A new time domain boundary element formulation and solution procedure for generalized dynamic coupled thermoelasticity is developed. Velocity and acceleration terms in the governig equations are replaced by finite difference type expressions according to the implicit time integration scheme of Wilson θ and the resulting static-like equations are written in integral form in conjuction with (a) the elastostatic and steady-state heat conduction fundamental solutions and (b) the frequency domain fundamental solution pair of coupled thermoelasticity. In both cases boundary and interior domain discretization are necessary. Both of these formulations result in the same matrix equation form and the solution is obtained by a stepwise marching in time procedure.

A new boundary element technique to transient non-linear low penetration problems of multiconductor systems

This paper describes a time domain boundary element formulation for calculation of eddy currents in a system of conductors of magnetic material due to the passage of short current pulses. The perturbation method is applied to derive the implicit time domain non-linear surface impedance boundary condition (SIBC) for the normal component of the magnetic field (magnetic induction) on the surface of the conductor of homogeneous isotropic magnetic material with an arbitrary BH-curve. Then the SIBC is coupled with the surface integral equation for the scalar magnetic potential given by the boundary integral equation method. The formulation developed is applicable to 3-D problems as well as 2-D problems. It Is shown that no computational difficulties arise compared to solving the problem under the assumption that the field does not penetrate into the body (static problem). A numerical example is considered.

Recent developments in dynamic stress analyses by time domain BEM

Time-stepping boundary element (BE) procedures have proved to be an effective tool in dynamic interaction analyses, especially if the effects of transient excitations are to be investigated. However, most time domain BE formulations are obtained by means of the so-called direct BE method. In the present study, indirect time-stepping procedures are developed and checked with respect to applicability to an effective interior stress wave analysis. Also, only a few papers deal with unilateral dynamic contact, and almost none with the case of time-dependent frictional slip. Thus, in order to study dynamic interaction effects as realistic as possible, the influence of the frictional resistance on the contact stresses and on the whole dynamic behaviour is investigated by a direct BE formulation.