Fast Boundary Element Research Articles

Isogeometric multilevel quadrature for forward and inverse random acoustic scattering

We study the numerical solution of forward and inverse time-harmonic acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that it suffices to know the deformation field’s expectation and covariance at the scatterer’s boundary to model the surface’s Karhunen–Loève expansion. Leveraging on the isogeometric framework, we employ multilevel quadrature methods to approximate quantities of interest such as the scattered wave’s expectation and variance. By computing the wave’s Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problems validate the proposed approach.

Three-dimensional preconditioned FM-IBEM solution to broadband-frequency seismic wave scattering in a layered sedimentary basin

A three-dimensional (3-D) fast multipole indirect boundary element method (FM-IBEM) combined with an inner-outer preconditioned generalized minimal residual (GMRES) method algorithm is proposed to solve the broadband-frequency seismic wave scattering problem in complex sites. This method is established in three steps: first, the scattering wave field is developed with the dynamic Green's function; second, the dynamic Green's functions are expanded to generate a traditional FM-IBEM framework; and finally, the preconditioning matrix and GMRES algorithm are introduced to pre-process the dynamic Green's functions. Using this method, the broadband-frequency scattering of normally incident seismic waves in a hemispherical layered sedimentary basin is investigated in the frequency and time domains. The results show that the proposed method is highly accurate and efficient for solving low- and high-frequency seismic wave problems. Compared with the FM-IBEM without preconditioning, the computational time and storage space required for the FM-IBEM with preconditioning are reduced by 84% and 54%, respectively. The layered sedimentary basin has significant wave amplification (of up to 10 times) for dimensionless frequencies, η, in the range of 0.5–3.0; however, for η > 5, the amplification effect becomes weak.

Differential algebraic fast multipole-accelerated boundary element method for nonlinear beam dynamics in arbitrary enclosures

A novel method is developed to take into account realistic boundary conditions in intense nonlinear beam dynamics. The algorithm consists of three main ingredients: the boundary element method that provides a solution for the discretized reformulation of the Poisson equation as boundary integrals; a novel fast multipole method developed for accurate and efficient computation of Coulomb potentials and forces; and differential algebraic methods, which form the numerical structures that enable and hold together the different components. The fast multipole method, without any modifications, also accelerates the solution of intertwining linear systems of equations for further efficiency enhancements. The resulting algorithm scales linearly with the number of particles $N$, as $m\text{ }\mathrm{log}\text{ }m$ with the number of boundary elements $m$, and, therefore, establishes an accurate and efficient method for intense beam dynamics simulations in arbitrary enclosures. Its performance is illustrated with three different cases and structures of practical interest.

Elaboration of the Fast Boundary Element Method for 3D Simulation of the Dynamics of a Bubble Cluster with Solid Particles in an Acoustic Field

Elaboration of the Fast Boundary Element Method for 3D Simulation of the Dynamics of a Bubble Cluster with Solid Particles in an Acoustic Field

Basic investigation to develop a fast time-domain boundary element method for electromagnetic scattering problems in 3D

Basic investigation to develop a fast time-domain boundary element method for electromagnetic scattering problems in 3D

A shape optimization for 3D unsteady acoustic problems using the fast time-domain boundary element method

A shape optimization for 3D unsteady acoustic problems using the fast time-domain boundary element method

Simulation of Adhesive Contact of Soft Microfibrils

Adhesive contact between a flat brush structure with deformable microfibrils and an elastic half space is numerically simulated. The stiffness of pillars is modeled by linear springs. The fast Fourier transform-assisted boundary element method for the contact of rigid indenters is modified to include the microfibril stiffness so that the deflection of pillars and elastic interaction to elastic foundation are coupled. In the limiting case of rigid pillars (pillar stiffness is much larger than the contact stiffness), the adhesive force is determined by the filling factor of brush, as described earlier. In the case of very soft pillars, the adhesive force is proportional to N1/4, where N is the number of pillars. The influence of relative stiffness, number and distribution of pillars on adhesive force is studied numerically. The results from both regularly and randomly distributed pillars show that the adhesive force is enhanced by splitting a compact punch into microfibrils and this effect becomes larger when the fibrils are softer.

Scattering of seismic waves by three-dimensional large-scale hill topography simulated by a fast parallel IBEM

To solve seismic wave scattering by a large-scale three-dimensional (3-D) hill topography, a fast parallel indirect boundary element method (IBEM) is developed by proposing a new construction method for the wave field, modifying the generalized minimum residual (GMRES) algorithm and constructing an OpenMP plus MPI parallel model. The validations of accuracy and efficiency show that this method can solve 3-D seismic response of a large-scale hill topography for broadband waves, and overcome the weakness of large storage and low efficiency of the traditional IBEM. Based on this new algorithm architecture, taking the broadband scattering of plane SV waves by a large-scale Gaussian-shaped hill of thousands-meters height as an example, the influence of several important parameters is investigated, including the incident frequency, the incident angle and the height-width and length-width ratio of the hill. The numerical results illustrate that the amplification effect on the ground motion by a near-hemispherical hill is more significant than the narrow hill. For low-frequency waves, the scattering effect of the higher hill is more pronounced, and there is only a single peak near the top of the hill. However, for high-frequency waves, rapid spatial variation of displacement amplitude appears on the hill surface.

A fast boundary element based solver for localized inelastic deformations

SummaryWe present a numerical method for the solution of nonlinear geomechanical problems involving localized deformation along shear bands and fractures. We leverage the boundary element method to solve for the quasi‐static elastic deformation of the medium while rigid‐plastic constitutive relations govern the behavior of displacement discontinuity (DD) segments capturing localized deformations. A fully implicit scheme is developed using a hierarchical approximation of the boundary element matrix. Combined with an adequate block preconditioner, this allows to tackle large problems via the use of an iterative solver for the solution of the tangent system. Several two‐dimensional examples of the initiation and growth of shear‐bands and tensile fractures illustrate the capabilities and accuracy of this technique. The method does not exhibit any mesh dependency associated with localization provided that (i) the softening length‐scale is resolved and (ii) the plane of localized deformations is discretized a priori using DD segments.

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

SummaryThree‐dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the ‐transform and the convolution quadrature method, we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency‐domain BEMs. Then, taking advantage of a well‐designed high‐frequency approximation, we drastically reduce the number of frequency‐domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regard to the time discretization and for the spacial discretization, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid‐structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity and the scattering of an abrupt wave by simple and realistic geometries.

Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

A numerical study of JKR-type adhesive contact of ellipsoids

The approximate solution by Johnson and Greenwood (2005 J. Phys. D: Appl. Phys. 38 1042–6) for an adhesive contact of an ellipsoid and an elastic half-space is revisited numerically using the fast Fourier transformation-based boundary element method. While for moderate values of the ratio of principal radii of the ellipsoid, R1/R2, predictions of the Johnson–Greenwood approximate theory are very good, they become increasingly inaccurate for large values of this parameter. On the basis of numerical simulations, we provide analytical expressions for the dependencies between load, approach and contact area and compare the exact shape of the contact area with the elliptical one assumed in the Johnson–Greenwood-theory.

On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic [formula omitted]-matrix accelerated Boundary Element Methods

This article is concerned with the derivation of fast Boundary Element Methods for 3D acoustic and elastodynamic problems. In particular, we are interested in the acceleration of Hierarchical matrix (H-matrix) based iterative solvers. While H-matrix representations allow to reduce the storage requirements and the cost of a matrix–vector product, the number of iterations for an iterative solver, as the frequency or the problem size increases, remains an issue.We consider an inner–outer preconditioning strategy, i.e., the preconditioner is applied through an iterative solver at the inner level. The preconditioner is defined as a H-matrix representation of the system matrix with a given accuracy. We investigate the influence of various parameters of the preconditioner, i.e., the H-matrix accuracy, the GMRES threshold and the maximum number of iterations of the inner solver. Different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system. Finally, we propose a way to define the optimal setting for this preconditioner.

On some recent developments of the 3D fast time-domain boundary element method

On some recent developments of the 3D fast time-domain boundary element method

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

This paper presents a grid-free simulation algorithm for the fully three-dimensional Vlasov--Poisson system for collisionless electron plasmas. We employ a standard particle method for the numerical approximation of the distribution function. Whereas the advection of the particles is grid-free by its very nature, the computation of the acceleration involves the solution of the nonlocal Poisson equation. To circumvent a volume mesh, we utilize the fast boundary element method, which reduces the three-dimensional Poisson equation to a system of linear equations on its two-dimensional boundary. This gives rise to fully populated matrices which are approximated by the $\mathcal H^2$-technique, reducing the computational time from quadratic to linear complexity. The approximation scheme based on interpolation has been shown to be robust and flexible, allowing a straightforward generalization to vector-valued functions. In particular, the Coulomb forces acting on the particles are computed in linear complexity. In first numerical tests, we validate our approach with the help of classical nonlinear plasma phenomena. Furthermore, we show that our method is able to simulate electron plasmas in complex three-dimensional domains with mixed boundary conditions in linear complexity.

A fast directional boundary element method for wideband multi-domain elastodynamic analysis

A fast directional boundary element method for wideband multi-domain elastodynamic analysis is developed. The directional low rank property of the elastodynamic kernel is proved, which serves as the theoretical basis of the fast directional algorithm. The boundary integral evaluation is accelerated domain by domain. For each domain, the octree is divided into low frequency and high frequency regimes according to its S-wave number. The computation in the low frequency regime is the same with the kernel independent fast multipole method, while the high frequency regime is computed efficiently by applying the parabolic low rank property of the kernels on directional wedges. The matrices on only one directional wedge for each level have to be calculated, since that for other directions can be obtained by coordinate frame rotations. Thus the harmonic responses for any frequencies can be computed efficiently. Numerical results show that the computational complexity for wideband multi-domain elastodynamic problems is successfully brought down to O(NlogαN). Its applications to viscoelastic materials and inverse problems are also validated.

Optimizations of a fast multipole symmetric Galerkin boundary element method code

This paper presents some optimizations of a fast multipole symmetric Galerkin boundary element method code. Except general optimizations, the code is specially sped up for crack propagation problems. Existing useful computational results are saved and re-used during the propagation. Some time-consuming phases of the code are accelerated by a shared memory parallelization. A new sparse matrix method is designed based on coordinate format and compressed sparse row format to limit the memory required during the matrix construction phase. The remarkable performance of the new code is shown through many simulations including large-scale problems.

Distributed fast boundary element methods for Helmholtz problems

We present an approach for a distributed memory parallelization of the boundary element method. The given mesh is decomposed into submeshes and the respective matrix blocks are distributed among computational nodes (processes). The distribution which takes care of the load balancing during the system matrix assembly and matrix-vector multiplication is based on the cyclic graph decomposition. Moreover, since the individual matrix blocks are approximated using the adaptive cross approximation method, we describe its modification capable of dealing with zero blocks in the double-layer operator matrix since these are usually problematic when using the original adaptive cross approximation algorithm. Convergence and scalability of the method are demonstrated on the half- and full-space sound scattering problems modeled by the Helmholtz equation.

Boundary element method for fracture mechanics analysis of 3d non-planar cracks in anisotropic solids

Among the numerical approaches to fracture mechanics analysis of cracked anisotropic solids, the boundary element method is notable for high accuracy and performance due to its semi-analytical nature and the use of only boundary mesh. Various boundary element techniques were proposed for 3D fracture mechanics analysis. However, the main problem of these approaches is the treatment of singular and hypersingular integrals, which can demand analytic evaluation of coefficient at kernel singularity at singular point in local curvilinear coordinates, which produce cumbersome equations in the case of non-planar geometries. Therefore, the paper presents novel formulation of the boundary element method for 3D fracture mechanics analysis of anisotropic solids with non-planar cracks. Pan’s single domain boundary element formulation is extended with several novelties, which allow accurate analysis of non-planar geometries. These are modified Kutt’s numerical quadratures with Chebyshev nodes for accurate evaluation of singular and hypersingular integrals; polynomial mappings for smoothing the integrand at the crack front line; and special shape functions, which account for a square-root stress singularity at crack front and allow accurate determination of stress intensity factors. The kernels of boundary integral equations are evaluated using the exponentially convergent quadrature, which allows derivation of fast boundary element technique. The procedure for accurate numerical determination of stress intensity factors at arbitrary point of the crack front is also developed. Numerical examples are presented, which show high accuracy of the proposed boundary element method. It is shown that non-planar cracks exhibit shearing mode opening along with normal one due to its geometry and direction of the applied loading. Present approach can be combined with Sih’s strain energy density criterion to study 3D cracks propagation in anisotropic elastic solids under fatigue loading, which is the direction of future research.

Fast direct isogeometric boundary element method for 3D potential problems based on HODLR matrix

A novel fast direct solver based on isogeometric boundary element method (IGABEM) is presented for solving 3D potential problems, which uses the hierarchical off-diagonal low-rank (HODLR) matrix structure arising from the discretization of boundary integral equations. Since the HODLR matrix can be factored into the product form of some diagonal blocks, we can use the Sherman–Morrison–Woodbury formula to solve the inverse of a HODLR matrix efficiently. For large scale problems, an accelerated adaptive cross approximation algorithm is developed to decompose the off-diagonal submatrices. In numerical implementation, bivariate NURBS basis functions are used to describe the geometry. Meanwhile, the same NURBS basis functions are also used to approximate the unknown boundary quantities. The present method is applied to some numerical examples, including an infinite space containing twenty spherical cavities. The numerical results clearly show that the fast direct solver developed in the paper can obtain accurate results with less CPU time.