Time-domain Boundary Element Method Research Articles

On a stabilization of the Ingard-Myers impedance boundary condition and its time domain implementation

It has been well-known that the Ingard-Myers impedance condition, while simple to apply, is subject to the hydrodynamic Kelvin-Helmholtz-type instability due to its use of a vortex sheet in modeling the flow at the liner boundary. Recently, in the development of a time domain boundary element method for acoustic scattering by treated surfaces, it was found that by neglecting a certain second-order spatial derivative term in the Ingard-Myers formulation, the hydrodynamic instability can be avoided. The present paper aims to provide further analysis of this modified condition, hereby referred to as the Truncated Ingard-Myers Impedance Boundary Condition (TIMIBC). It will be shown, based on the dispersion relations of linear waves, that the instability intrinsic to the Ingard-Myers condition is eliminated in the proposed new formulation. Quantitative assessments on the accuracy of the TIMIBC for scattering of acoustic waves by lined surfaces are carried out, and its effectiveness is demonstrated by a numerical example. It is found that the TIMIBC provides a good approximation to the original Ingard-Myers condition for flows of low to mid subsonic Mach numbers. As such, the proposed TIMIBC can offer a practical solution for overcoming the intrinsic instability associated with the Ingard-Myers condition. Moreover, time domain implementation of the TIMIBC is also discussed and illustrated with a numerical example using a finite difference scheme. In particular, a minimization procedure for finding the poles and coefficients of a broadband multipole expansion for the impedance function is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured.

Convolution quadrature time-domain boundary element method for antiplane anisotropic viscoelastic wave propagation

This paper presents a Convolution Quadrature Time-Domain Boundary Element Method (CQBEM) for antiplane anisotropic viscoelastodynamics. The proposed CQBEM formulation employs the standard linear viscoelastic model and the fundamental solution developed by Wang and Achenbach for expressing viscoelastic and anisotropic properties, respectively. This fundamental solution, which involves integration over the unit circle, is evaluated numerically. After offering a comprehensive overview of the proposed CQBEM formulation, we demonstrate the effectiveness of the proposed method by analyzing the scattering of elastic waves by cavities in certain types of anisotropic viscoelastic materials. Additionally, the use of group velocity curves further validates the numerical results, confirming the anisotropic and viscoelastic effects and thus substantiates the proposed CQBEM for antiplane anisotropic viscoelastic wave propagation.

A simple TD-BEM model for heterogeneous orthotropic hill-shaped topographies

A simple numerical model named DASBEM was successfully developed to analyze the seismic heterogeneous orthotropic hill-shaped topographies by a time-domain boundary element method (TD-BEM) based on half-space Green’s functions. The model was elaborated only by discretizing the hill surface and its interface with the underlying half-space through the use of image source theory and the substructure approach. To improve the model at the corners, the double node procedure was applied to the extreme nodes of the quadratic elements. An attenuation ratio is implemented in the boundary equations using a decremental exponential function. After presenting the technique, a validation example is presented alongside the literature to measure the convergence with an isotropic response. Next, a sample Gaussian-shaped hill model is prepared under propagating obliquely incident SH-waves as a common sample topography and the surface response is obtained by considering some significant parameters as well as the shape ratio, isotropy factor, frequency content, and angle of the incident wave. The ground surface response is sensitized in two time and frequency domains. The results showed that the amplitude of the response was not only dependent on the impedance ratio but also the orthotropy ratio, which was always effective in orienting the wave-front to amplify the ground movement.Graphical

A time-domain boundary element method using a kernel-function library for 3D acoustic problems

Time-domain boundary element method (TDBEM) can be applied in studying transient acoustic wave problems, especially in solving exterior problems. However, in order to avoid calculations of coefficients at different time steps, TDBEM usually requires storing or recomputing the coefficients at each time step. This can lead to significant amount of memory usage or long computing time. In this paper, an acoustic TDBEM based on a kernel-function library (KFL-BEM) is proposed, in order to reduce the memory consumption of the time-domain conventional BEM (CBEM) and speedup the computation. In this approach, the storage requirement is reduced from O(N2Ntmin) to O(N2), where N represents the number of degrees of freedom of the model, and Ntmin is the minimum number of time steps for which coefficients need to be computed and stored. To demonstrate the effectiveness of the KFL-BEM, two verification examples are presented using the problems of a pulsating sphere and sound propagating in a channel. Compared with the CBEM, the KFL-BEM can save significantly the memory storage as it does not require storing coefficients for any previous time steps. This method can also be applied to solve vibro-acoustic problems in the time domain. As an example, the acoustic radiation responses of a tuning fork under different striking loads are studied using the finite element method and the proposed KFL-BEM, which clearly shows the potentials of the KFL-BEM in solving time-domain acoustic problems.

Validation of a panel method with full-scale FOWT measurements and verification with engineering models

The present paper gives insights into the verification of different models of the Floatgen demonstrator by BW Ideol. This floating offshore wind turbine (FOWT) is based on a barge type-floater with moonpool. The moonpool shall dampen the platform’s motion and is challenging to model due to the increased complexity of the hydrodynamic behaviour. For the verification purposes, the results of two engineering methods (OrcaFlex, OpenFAST) equipped with frequency-dependent hydrodynamic solver and the nonlinear time-domain boundary element method panMARE are compared. The OrcaFlex model designed by BW Ideol is regarded as a reference because of a preceding calibration with measurement data. All models show fairly good agreement in most cases. Only in heave there are significant differences between engineering and the nonlinear models which is caused by different modelling of the moonpool hydrodynamics. In addition, a hybrid simulation approach with prescribed motion and constrained wind field based on motion sensors and LIDAR measurements on the Floatgen demonstrator was applied for an aerodynamic and load validation of panMARE. The validation shows good agreement based on time-domain comparison regarding the tower top loads and turbine operation.

Realizations of the generalized adaptive cross approximation in an acoustic time domain boundary element method

AbstractIn acoustics, the boundary element method (BEM) is much more common compared to elasticity. This is driven by the applications, which are in acoustics very often radiation problems causing trouble in finite element or finite volume methods. Nevertheless, for a time domain calculation, still an efficient BE formulations is lacking. We consider the time domain BEM for the homogeneous wave equation with vanishing initial conditions and given Dirichlet boundary conditions. The generalized convolution quadrature method (gCQ) developed by Lopez‐Fernandez and Sauter is used for the temporal discretisation. The spatial discretisation is done classically. Essentially, the gCQ requires to establish boundary element matrices of the corresponding elliptic problem in Laplace domain at several complex frequencies. Consequently, an array of system matrices is obtained. This array of system matrices can be interpreted as a three‐dimensional array of data, which should be approximated by a data‐sparse representation. The adaptive cross approximation (ACA) can be generalized to handle these three‐dimensional data arrays. Adaptively, the rank of the three‐dimensional data array is increased until a prescribed accuracy is obtained. On a pure algebraic level, it is decided whether a low‐rank approximation of the three‐dimensional data array is close enough to the original matrix. Hierarchical matrices are used in the two spatial dimensions and the third dimension are the complex frequencies. Hence, the algorithm makes not only a data sparse approximation in the two spatial dimensions but detects adaptively how much frequencies are necessary for which matrix block.

A time‐domain boundary element method for the 3D dissipative wave equation: Case of Neumann problems

AbstractThe present article proposes a time‐domain boundary element method (TDBEM) for the three‐dimensional (3D) dissipative wave equation (DWE). Although the fundamental ingredients such as the Green's function for the 3D DWE have been known for a long time, the details of formulation and implementation for such a 3D TDBEM have been unreported yet to the author's best knowledge. The present formulation is performed truly in time domain on the basis of the time‐dependent Green's function and results in a marching‐on‐in‐time fashion. The main concern is in the evaluation of the boundary integrals. For this regard, weakly‐ and removable‐singularities are carefully treated. The proposed TDBEM is checked through the numerical examples whose solutions can be obtained semi‐analytically by means of the inverse Laplace transform. The results of the present TDBEM are satisfactory to validate its formulation and implementation for Neumann problems. On the other hand, the present formulation based on the ordinary boundary integral equation (BIE) is unstable for Dirichlet problems. The numerical analyses for the non‐dissipative case imply that the instability issue can be partially resolved by using the Burton–Miller BIE even in the dissipative case.

A fast time-domain boundary element method for three-dimensional electromagnetic scattering problems

This paper proposes a fast time-domain boundary element method (TDBEM) to solve three-dimensional transient electromagnetic scattering problems regarding perfectly electric conductors in the classical marching-on-in-time manner. The algorithm of the fast TDBEM is a time-domain variant of the interpolation-based fast multipole method (IFMM), which is similar to the time-domain IFMM for acoustic scattering problems investigated in the author's previous studies. The principle of the present IFMM is to interpolate the kernel functions of the electric and magnetic field integral equations (EFIE and MFIE, respectively) so that every kernel function is expressed in a form of separation of variables in terms of both the spatial and temporal variables. Such an expression enables to construct a fast method to evaluate the scalar and vector potentials in the EFIE and MFIE with using so-called multipole-moments and local-coefficients associated with a space-time hierarchy. As opposed to $O(N_s^2 N_t)$ of the conventional TDBEM, the computational complexity of the fast TDBEM is estimated as $O(N_s^{1+\delta}N_t)$, where $N_s$ and $N_t$ stand for the spatial and temporal degrees of freedom, respectively, and $\delta$ is typically $1/2$ or $1/3$. The numerical examples presented the advantages of the proposed fast TDBEM over the conventional TDBEM when solving large-scale problems.

Higher-order time domain boundary elements for elastodynamics: graded meshes and hp versions

The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasi-optimal estimates for piecewise polynomial approximations of the Dirichlet trace of the solution and the traction. The results are applied to hp and graded versions of the time domain boundary element method for the weakly singular and the hypersingular integral equations. Numerical examples confirm the theoretical results for the Dirichlet and Neumann problems for screens and for polygonal domains in 2d. They exhibit the expected quasi-optimal convergence rates and the singular behavior of the solutions.

A half-space TD-BEM model for a seismic corrugated orthotropic stratum

In this paper, the model of a linear corrugated orthotropic stratum above a semi-infinite substratum is elaborated by a half-space time-domain boundary element method (TD-BEM) subjected to oblique incident plane SH waves. The ground surface and the interface are corrugated in a similar ratio using a Gaussian-shaped function. In the used approach, only the corrugated boundaries are required to be discretized. An incremental mesh scheme is applied to reduce the number of boundary elements in the large width of the interface. After developing the algorithm and solving a verification example, the seismic ground response is obtained and compared by assuming three cases of the model based on the isotropy factor, one isotropic case, and two orthotropic cases. Some effective parameters, including the shape ratio of the corrugation, frequency content, and wavefront propagation angle, are also considered to obtain the amplification pattern of the surface. Finally, by developing the method to multi-layered medium, its efficiency is investigated in the analysis of orthotropic heterogeneous models with more degrees of freedom. The results showed that both geometry and material factors were directly affected by the phenomenon of wave propagation so that the corrugation depth and the velocity of the wavefront in different directions could significantly control the response of the ground surface.

The Effect of a Linear Free Surface Boundary Condition on the Steady-State Wave-Making of Shallowly Submerged Underwater Vehicles

Near-surface simulation methods for shallowly submerged underwater vehicles are necessary for the population of a variety of free-surface-affected, coefficient-based maneuvering and seakeeping models. Simulations vary in complexity and computational costs, often sacrificing accuracy for simplicity and speed. One particular simplifying assumption, the linearization of the free surface boundary conditions, is explored in this study by comparing the steady-state wave-making characteristics of a shallowly submerged prolate spheroid using two different simulation methods at several submergence depths and forward speeds. Hydrodynamic responses are compared between a time-domain boundary element method that makes use of a linearized free surface boundary condition and an inviscid, volume of fluid Reynolds-Averaged Navier–Stokes computational fluid dynamics code that imposes no explicit free surface boundary condition. Differences of up to 22.6%, 32.5%, and 33.3% are found in the prediction of steady state surge force, heave force, and pitch moment, respectively. The largest differences between the two simulation methods arise for motions occurring at small submergences and large wave-making velocities where linear free-surface assumptions become less valid. Nonlinearities that occur in such cases are revealed through physical artifacts such as wave steepening, wave breaking, and high-energy waves. A further examination of near-surface viscous forces reveals that the viscous drag on the vessel is depth dependent due to the changing velocity profile around the body.

A free surface corrected lumped parameter model for near-surface horizontal maneuvers of underwater vehicles in waves

We provide theory and results obtained from the application of a 6-degree of freedom lumped parameter maneuvering model (LPMM) able to predict the maneuvering motions of underwater vehicles navigating at shallow depth below free surface waves. The parameters of the maneuvering model are identified using a combination of steady and unsteady captive maneuvers, simulated with high fidelity computational fluid dynamic (CFD) methods. This work focuses on correcting the LPMM, which is accurate for deep water motions, to account for free surface effects. A frequency domain strip theory method is used to account for changes in the added mass due to free surface proximity, to calculate memory forces, and to estimate wave excitation forces while a 3D time domain boundary element method is used to predict steady-state wave making forces. The result is a combined maneuvering and seakeeping model for underwater vehicles operating at shallow depths below the free surface. Near-surface horizontal zig-zag motion predictions in both calm water and under waves reveal the importance of including the free surface as vehicle trajectories at shallow depths differ substantially from those at deeper submergences for identical maneuvering inputs.

Numerical study on the water entry of two-dimensional airfoils by BEM

Hydrodynamics of the traditional wedges, cylinders, and cones entering water have been heavily studied. However, the hydrodynamic characteristics of airfoils are practically unknown; the problem is inspired by the recently widely developed fixed-wing unmanned aerial-underwater vehicle, which can fly in the air, cruise in water and frequently enters and exits free water surface. In this study, the hydrodynamic characteristics and fully nonlinear free surface deformation for the water entry of airfoils have been investigated by the time domain boundary element method. First, convergence study based on the NACA0012 airfoil is considered. The time step and element size are adjusted, and the results show that the free surface elevation and pressure distribution agree well with those obtained through Reynolds-averaged Navier–Stokes equations. The results show that the higher the entry velocity, the greater the peak hydrodynamic force. Furthermore, the higher thickness ratio causes greater added mass. Gravity and velocity are found to significantly influence the hydrodynamic behavior, including the time-varying force, while thickness ratio has a great impact on added mass and pressure.

Evaluation of the Stability of Time Domain Boundary Element Method in Seismic Analysis of Heterogeneous Environments

Evaluation of the Stability of Time Domain Boundary Element Method in Seismic Analysis of Heterogeneous Environments

A development of sequential estimation method for position and shape of acoustic scatterers by the time-domain boundary element method

A development of sequential estimation method for position and shape of acoustic scatterers by the time-domain boundary element method

Time domain boundary element method for semi-infinite domain problems using CSR storage method

The time domain boundary element method (TDBEM) is suitable for dealing with dynamic problems in the semi-infinite domain (such as the propagation of seismic waves). The fundamental solution in the TDBEM is an impulse function with time terms, and the formed coefficient matrix is sparse. In this paper, a TDBEM using the compressed storage algorithm based on Compressed Sparse Row (CSR) format is proposed to solve the semi-infinite domain dynamics problems. The coefficient matrix is stored in the CSR format, and the generation method of the coefficient matrix elements and the matrix operation scheme based on CSR format in the TDBEM are given. The GMRES algorithm is also modified to make it suitable for TDBEM of CSR format to improve the efficiency of iterative solution. The provided semi-infinite domain dynamics examples show that the TDBEM of CSR format proposed in this paper can greatly reduce the storage space and improve the efficiency and scale of the solution.

A study on singular boundary integrals and stability of 3D time domain boundary element method

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.

Corner treatment in 3D time-domain boundary element method

Corner treatment in 3D time-domain boundary element method

A multiscale method coupling peridynamic and boundary element models for dynamic problems

A novel multiscale method coupling peridynamics (PD) and boundary element method (BEM) is proposed for dynamic problems. In the coupling model, the discontinuous behaviors of materials such as crack development and branching processes are simulated by PD while the boundary and infinite domain are modeled by time-domain BEM, which can take full advantage of their salient features. A general stiffness matrix is introduced in the coupling model to combine the PD and BEM governing equations in such a way that the continuity of displacement and the equilibrium of tractions at the PD-BEM interface are satisfied. A variable horizon method is developed to further eliminate boundary effects and ghost forces at the coupling interface between PD and BEM. Then, a concurrent coupling procedure of PD and BEM is established and its implementation is also introduced based on the dynamic explicit algorithm, i.e. the central difference method. The proposed coupling method is verified by comparing its predictions with those from analytical and experimental results. Numerical examples demonstrate the applicability of the proposed model for wave propagation, infinite domain and fracture problems under dynamic loads.

A time domain boundary element method for acoustic scattering by lined surfaces in a subsonic uniform mean flow

A time domain boundary integral equation with Burton-Miller reformulation is presented for acoustic scattering by surfaces with liners in a uniform mean flow. The Ingard-Myers impedance boundary condition is implemented using a broadband multipole impedance model which is in turn converted into time domain differential equations to augment the boundary integral equation. The coupled integral-differential equations are solved numerically by a March-On-in-Time (MOT) scheme. While the Ingard-Myers condition is known to support Kelvin-Helmholtz instability due to its use of a vortex sheet interface between the flow and the lined surface, it is found that by neglecting a second-order derivative term in the current time domain impedance boundary condition formulation, the instability can be effectively avoided in computation. Proposed formulation and implementation are validated with numerical examples. Moreover, a minimization procedure for finding the poles and coefficients of the broadband multiple impedance expansion is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured. Numerical tests show the proposed minimization approach is effective for modeling liners that are commonly used in aeroacoustic applications.