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.

Efficient calculation of the three-dimensional sound pressure field around a slab track

The wavenumber domain Boundary Element (2.5D BE) method is well suited to calculate the acoustic sound field around structures with a constant cross-section along one dimension, such as noise barriers or railway track. By expressing the sound field along this dimension in wavenumber domain, the numerical model is reduced from a 3D model to 2D model at each wavenumber. A consequence of the required discrete Fourier domain representation is that the sound field is represented by periodically repeating sections, of which only one section is physically meaningful. The resolution and the number of required wavenumbers increases with the desired length and spatial discretisation of this section. Describing the sound field adequately to auralise the sound without disturbing artefacts requires a large number of wavenumbers (and thus 2D BE computations), which is not feasible for large geometries. Here, a method is introduced that allows the calculation of the 3D sound field by solving a single 2D BE problem for a dense frequency spectrum and interpolating at higher wavenumbers. The calculation efficiency is further increased by precalculating the acoustic transfer functions between each BE surface element and receiver positions. Combining these two methods allows for efficient calculation of the 3D sound field around acoustically rigid structures such as slab tracks. The numerical approach is validated by comparison with a standard 2.5D BEM calculation and an analytical solution. Precalculated transfer functions to calculate the sound radiation from railway track, which are made available online, are illustrated. An example application is presented.

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.

The sound radiation from a periodic array of railway sleepers using a wavenumber domain method

Rolling noise, radiated by the vibration of the wheels and the track, is an important source of noise from railway operations. In ballasted track the sleepers supporting the rails form an important source of noise at low frequencies. There are difficulties to calculate their sound radiation using numerical models due to their discrete periodic nature and the infinite extent of the track. For the rail, which has invariant geometry in the axial direction, the wavenumber domain (2.5D) finite element or boundary element method can be used to calculate the vibration and noise radiation. However, the 2.5D method cannot be used directly to predict the vibration and sound radiation of railway sleepers due to their discrete spacing. In this work, the discrete spacing of the sleepers in the spatial domain is introduced into the 2.5D method by applying a series of rectangular windows according to the sleeper width and spacing. The vibration of the sleepers in the wavenumber domain is obtained by applying a spatial Fourier transform to the product of these windows and the rail transfer mobility, also allowing for the ratio between the sleeper vibration and the rail vibration. The sleeper radiation obtained from the proposed approach is compared with the result obtained from the Rayleigh integral method, showing good agreement. The mobilities of the sleepers in the wavenumber domain are used to explain the effect of the sleeper spacing on the sound radiation. The spectrum of response in the wavenumber domain contains multiple waves, shifted in wavenumber by multiples of 2π/L, where L is the sleeper spacing. At low frequency, only the fundamental wave corresponding to the rail wavenumber radiates power to the far field; the ratio of the sound power of discrete sleepers to that of continuous sleepers is shown to be equal to the square of the width-to-spacing ratio. As the frequency increases, new wave branches enter the acoustic wavenumber range, leading to peaks in the radiation efficiency. At high frequency, the ratio of the sound powers converges to a constant value equal to the ratio of areas.

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 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.

A local domain boundary element method for solving 2D incompressible fluid flow problems

The Boundary Element Method (BEM) is a well-established, accurate numerical method for solving engineering problems described by linear partial differential equations. However, the application of BEM to non-linear problems leads to the appearance of volume integrals in the integral formulation, and combined with the non-local character of the method the required computational effort is dramatically increased. In this work, a Local Domain BEM (LD-BEM) is proposed for solving incompressible fluid flow problems. The domain of interest is fragmented into small subdomains, in each of which the integral representation of the solution is considered separately. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and reduced computational complexity, overcoming some of the well-known disadvantages of the conventional BEM. The proposed method is applied to the solution of the well-known two-dimensional (2D) lid-driven cavity problem, for Reynolds numbers up to Re=125 × 103, with a discretization involving more than 10 million unknowns.

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

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.

Three-dimensional elastic analysis of a bi-material system with a single domain boundary element method

The three dimensional fundamental solution of bi-materials is introduced to the boundary element method (BEM) for elastic analysis of a bi-material system instead of solving a multi-domain problem. Using the explicit form of the Green’s function for bi-materials, the boundary integral equations (BIE) are set up for a finite bi-layered material system with a plane interface. The Green’s function for a bi-material system combines the Kelvin’s solution and disturbed fields by image source. Because it exactly satisfies the jointed continuity conditions at the interface (x3=0), no internal mesh is required along the interface and a single domain BEM can be established. The potential numerical errors of the multi-domain BEM using the Kelvin’s solution due to the mesh quality at the interface can be avoided. This single domain BEM for bi-material saves the effort of domain discretization, predicts the singularity on the interface analytically, and enables an efficient, straightforward analysis similar to the homogeneous solids. It is particularly suitable to simulate thin film/substrate systems, overlay/base structures, and other similar layered materials. The numerical simulations verify the effectiveness and convergence of the method, and the case studies of wind turbine blades and solar panels demonstrate the industrial applications of this method.

Computational elastodynamics of functionally graded thick-walled cylinders and annular coatings subjected to pressure shocks

A computational technique based on domain-boundary element method (D-BEM) is developed for elastodynamic analysis of functionally graded thick-walled cylinders and annular coatings subjected to pressure shock type of loadings. The formulation is built on the wave equation, which is derived in accordance with plane elastodynamics. Weighted residual statement for the wave equation is expressed by using the static fundamental solution as the weight function. Applying integration by parts and incorporating the boundary conditions, the problem is reduced to an integral equation. Problem domain is discretized by quadratic cells to transform the integral equation into a system of ordinary differential equations in time. Equation system is solved numerically applying Houbolt's method. Developed procedure is verified through comparisons to the analytical results available in the literature. Parametric analyses are carried out considering short-time ramp and exponential variation types of pressure shocks. Presented numerical results illustrate the influence of material property gradation on time histories and spatial distributions of displacement and stress components in FGM thick-walled cylinders and annular coatings.

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.

Attenuation of wave force on a floating dock by multiple porous breakwaters

The impact of an array of porous breakwaters for reducing wave forces on a floating dock is studied based on the small-amplitude water wave theory in finite water depth. The breakwater is approximated by a porous bottom-standing rectangular structure and an impermeable floating rectangular structure is placed at a finite distance on the lee-side representing a dock. The physical problem is solved analytically using Eigenfunction expansion method and numerically using a hybrid element multi domain boundary element method. Two scenarios are considered: (i) wave scattering by N porous structures and the floating dock and (ii) wave reflection by a dock attached to the impermeable wall in presence of N porous structures. The reflection, transmission and dissipation coefficients are evaluated and analyzed for various structural and wave parameters. The study reveals that the Bragg reflection occurs in the case of wave scattering by an array of porous structures in the absence of floating dock. It is also found that in the absence of dock N−2 sub-harmonic peaks occurs whereas, in the presence of dock the effect of Bragg resonance is clearly evident on hydrodynamic coefficients. Also, the study concludes that for four structures with larger height, 100% of wave energy can be dissipated and thus mitigate the wave force on the floating rigid dock. Furthermore, the height and width of the porous structure and spacing between the structures play a vital role to protect the floating platform.

Laplace domain BEM for anisotropic transient elastodynamics

In this paper, we describe Laplace domain boundary element method (BEM) for transient dynamic problems of three-dimensional finite homogeneous anisotropic linearly elastic solids. The employed boundary integral equations for displacements are regularized using the static traction fundamental solution. Modified integral expressions for the dynamic parts of anisotropic fundamental solutions and their first derivatives are obtained. Boundary elements with mixed approximation of geometry and field variables with the standard nodal collocation procedure are used for spatial discretization. In order to obtain time-domain solutions, the classic Durbin’s method is applied for numerical inversion of Laplace transform. Problem of alleviating Gibbs oscillations is addressed. Dynamic boundary element analysis of the model problem involving trigonal material is performed to test presented formulation. Obtained results are compared with finite element solutions.

A local domain boundary element method for solving the nonlinear fisher KPP diffusion-reaction equation

The nonlinear Fisher-KPP diffusion-reaction equation has been successfully employed to model competitive processes in biological populations. A robust numerical method for solving diffusion problems is the Boundary Element Method (ΒΕΜ). However, the conventional BEM suffers from large CPU memory requirements since it leads to dense coefficient matrices and for nonlinear problems, its efficiency is further reduced due to the appearance of volume integrals. In this work, a Local Domain BEM (LD-BEM) for solving the Fisher KPP equation is presented. The domain of interest is fragmented into small subdomains and the integral representation of the problem is considered separately for each of the considered subdomains. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and a reduced number of degrees of freedom. Five numerical examples are solved to assess the accuracy of the proposed method.

Time-Domain Boundary Element Method with Broadband Impedance Boundary Condition

Acoustic liners are an effective tool for noise reduction and are characterized by a frequency-dependent impedance value. In this paper, a time-domain boundary element method for acoustic scattering coupled with a broadband impedance boundary condition is studied for the case of no mean flow. A Burton-Miller reformulation of the time domain boundary element method for acoustic scattering is carried out with both the pressure and its surface normal derivative terms retained. A multipole impedance model is converted into the time domain and used for enforcing an impedance boundary condition over a wide range of frequencies. Discretization of the coupled system is described. In particular, the time-domain impedance boundary condition is discretized by the third-order implicit backward difference scheme. Stability of the coupling is assessed by an eigenvalue analysis. Scattering solutions that demonstrate the validity and stability of the numerical method are presented.

Site Amplification Response by a Dual Reciprocity Boundary Element Method

ABSTRACT In this paper, a time domain boundary element method is developed for obtaining the seismic amplification of the surface topographies. Initially, Fourier series is applied into the dual reciprocity method and required boundary element fundamental solutions are extracted. After having presented the required formulations, accuracy of the method is verified by comparing its results with available analytical solutions. Finally, the topography of Zanjan-valley region is evaluated under the propagation of the P waves with different incident angles and frequencies. The results indicate that the variation of the seismic responses strongly depends on the wave properties and topology of the model.

Transient dynamic analysis of functionally graded micro-beams considering small-scale effects

A domain-boundary element method, based on modified couple stress theory, is developed for transient dynamic analysis of functionally graded micro-beams. Incorporating static fundamental solutions as weight functions in weighted residual expressions, governing partial differential equations of motion are converted to a set of coupled integral equations. A system of ordinary differential equations in time is obtained by domain discretization and solved using the Houbolt time marching scheme. Developed procedures are verified through comparisons to the results available in the literature for micro- and macro-scale beams. Numerical results illustrate elasto-dynamic responses of graded micro-beams subjected to various loading types. It is shown that metal-rich micro-beams and those with a smaller length scale parameter ratio undergo higher displacements and are subjected to larger normal stresses.