Efficient Boundary Element Research Articles

Boundary-Element Analysis of the Noise Scattering for Urban Aerial Mobility Vehicles: Solver Development and Assessment

Abstract Urban aerial mobility (UAM) vehicles with multipropeller configurations have attracted considerable attention in recent years. However, previous investigations have mainly focused on the aerodynamic noise from individual propellers, with limited focus on the fuselage’s sound scattering effects which can alter the far-field noise directivity. In this work, an efficient boundary element solver for sound scattering was developed to fill this gap. The solver employs hierarchical matrices to save computational cost. The benchmark examples showed high accuracy and good scalability. A representative vehicle model was then chosen, and the propeller noise was simulated using rotating sources. Results show that the shielding effect in the fuselage/propeller configuration can produce an apparent noise reduction and redirect sound energy distribution, suggesting the importance of considering the fuselage in low-noise UAM development.

Krauklis wave propagation within a complex fracture system: Modeling via a two-dimensional time-harmonic boundary element method.

Fluid-filled fractures involving kinks and branches result in complex interactions between Krauklis waves-highly dispersive and attenuating pressure waves within the fracture-and the body waves in the surrounding medium. For studying these interactions, we introduce an efficient 2D time-harmonic elastodynamic boundary element method. Instead of modeling the domain within a fracture as a finite-thickness fluid layer, this method employs zero-thickness, poroelastic Linear-Slip Interfaces to model the low-frequency, local fluid-solid interaction. Using this method, the scattering of Krauklis waves by a single kink along a straight fracture and the radiation of body waves generated by Krauklis waves within complex fracture systems are examined.

A combined displacement discontinuity-interaction integral method for computing stress intensity factors and T-stress

In this paper, the displacement discontinuity method (DDM) is combined with interaction integral to simultaneously evaluate stress intensity factors and T-stress for analyzing two-dimensional mixed-mode crack problems. The displacement discontinuity method has been proved to be one of the most efficient boundary element methods for solving crack problems with boundary-only discretization character. However, there is a lack of efficient methods collaborating with the displacement discontinuity method to evaluate fracture parameters. The available geometrical extrapolation method and J-integral technique for calculating fracture parameters in combination with the displacement discontinuity method are imprecise and difficult to be implemented into mixed-mode crack analysis. The present combined displacement discontinuity method and interaction integral approach can easily extract stress intensity factors and T-stress for mixed-mode crack problems without needing any decomposition of elastic field into symmetric and antisymmetric components. The fundamental basis lies in introducing auxiliary fields for the proper defined single mode crack, then calculating fracture parameters by evaluating the formulated interaction integral in terms of displacement discontinuity solutions and auxiliary elastic fields. In this paper, the basis of the displacement discontinuity method is illustrated firstly, then the explicit expressions for auxiliary fields of displacement gradients are derived. Next, the interaction integral can be determined by being converted to an equivalent area integral. Finally, several numerical examples are examined to demonstrate the correctness of the present method.

Dynamic BEM analysis of elastic anisotropic nano‐sheet with surface imperfections and cavities

AbstractThe paper considers elastic anisotropic nano‐sheet with nanotopography and containing embedded nanocavities under time‐harmonic in‐plane wave motion. The mechanical model is based on the classical elastodynamic theory for the anisotropic bulk solid and the nonclassical boundary conditions along its boundary derived by Gurtin and Murdoch. A localized constitutive equation for the half‐plane boundary is considered in the frame of surface elasticity theory. The computational tool is an efficient direct displacement boundary element method (BEM) based on the frequency‐dependent elastodynamic fundamental solution for elastic generally anisotropic material. To show the versatility of the proposed methodology, wave propagation in an elastic anisotropic heterogeneous nano‐sheet with nanorelief presented by hill‐canyon nanotopography is studied. The simulations illustrate the dependence of the wave field on the material anisotropy, on the surface elasticity properties, on the nanorelief peculiarities, on the nanocavities existence and their dynamic interaction, and on the incident wave characteristics. The obtained results reveal the potential of the developed mechanical model based on the boundary integral equations in the frame of surface elasticity theory to produce highly accurate results by using strongly reduced discretization mesh in comparison with the domain‐based methods.

Robust and efficient non-singular boundary element method for scattering and vibration with elastic waves

The propagation and scattering phenomena of linear elastic waves are often very complex. A robust and efficient numerical tool is required to study such dynamic linear wave phenomena. In this work, a fully three dimensional desingularized boundary element method is presented. Special chosen singular integrals are subtracted from the original integrals, thereby eliminating their (hyper) singular behavior. Validation of the numerical results is done with known analytical results. Then several examples on scattering and manipulation of linear elastic waves are shown for a phased array of vibrating pillars and for waves scattering on a rigid bowl.

Efficient spectral coupled boundary element method for fully nonlinear wave–structure interaction simulation

Accurately analyzing wave–structure interactions is crucial for the design and operational safety of ships and marine structures. This paper presents a fully nonlinear potential-flow approach for simulating wave–structure interactions using the newly proposed spectral coupled boundary element method (SCBEM). The SCBEM efficiently models an extensive water body that encompasses structures by establishing a boundary element method (BEM) computational domain solely around the object of interest while accurately simulating the far-field broad water by a spectral layer. To further improve efficiency, graphics processing unit acceleration is hired during iterative solving of the boundary value problem in the already small-sized interior BEM domain. Simulations are conducted to validate the accuracy of the method on cases with strong nonlinear phenomena, including wave run-up on a single cylinder, diffraction of a four-cylinder array, near-trapped modes for closely spaced columns, and gap resonance that occurred in side-by-side offloading. The wave run-up, diffraction wave pattern, near-trapped mode, and gap resonance frequency obtained by the proposed method are in good agreement with data from experiments and published literature. The quite good accuracy and the exceptional computational efficiency of the SCBEM demonstrate its promising potential for more application in practical marine problems.

Pixel-based boundary element method for computing effective thermal conductivity of heterogeneous materials

In the present article, we propose an efficient 2D Pixel-based Boundary Element formulation to compute the effective thermal conductivity of heterogeneous materials. In the proposed formulation, each pixel of the digital image is represented by a subdomain with four boundary elements. We develop the formulation within the context of the Symmetric Galerkin Boundary Element Method and, apart from the Boundary Element Method itself, the other main ingredients are the Domain Decomposition technique, to treat each pixel as a subdomain, and the Element-by-Element technique used in conjunction with the Preconditioned Conjugate Gradient method to solve the resulting symmetric positive definite large linear system. In addition, Periodic Boundary Conditions are employed. We validate our formulation and implementation with the aid of analytical results for the case of a plate with periodic circular inclusions. Further, we illustrate the application of the proposed formulation by computing the effective thermal conductivity of a plate with periodic square inclusions, a woven-like material, and a cast iron sample, and compare the obtained results with those obtained via finite elements. Our results indicate that the Pixel-based Boundary Element formulation performs, at least, as good as the Pixel-based Finite Element formulation.

Frequency Extraction for BEM Matrices Arising From the 3D Scalar Helmholtz Equation

The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and H-matrix based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an H-matrix. We show that the frequency dependency of this H-matrix can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine.

Coupling Heat Conduction and Radiation by an Isogeometric Boundary Element Method in 2-D Structures

We propose an efficient isogeometric boundary element method to address the coupling of heat conduction and radiation in homogeneous or inhomogeneous materials. The isogeometric boundary element method is used to construct irregular 2D models, which eliminate errors in model construction. The physical unknowns in the governing equations for heat conduction and radiation are discretized using an interpolation approximation, and the integral equations are finally solved by Newton–Raphson iteration; it is noteworthy that we use the radial integration method to convert the domain integrals to boundary integrals, and we combine the numerical schemes for heat conduction and radiation. The results of the three numerical cases show that the adopted algorithm can improve the computational accuracy and efficiency.

A New Approach to Space-Time Boundary Integral Equations for the Wave Equation

We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with current time dependent boundary element method's implementations and we predict that it explains many of the behaviours observed in practice but that were not understood with the existing theory.

Boundary element modeling of fractional nonlinear generalized photothermal stress wave propagation problems in FG anisotropic smart semiconductors

The main aim of this article is to develop an efficient boundary element method (BEM) modeling of the fractional nonlinear generalized photo thermal stress wave propagation problems in the context of functionally graded (FG) anisotropic smart semiconductors. Due to nonlinearity, fractional order heat conduction and strongly anisotropy of mechanical properties, the governing equations system of such problems is often very difficult to solve using classical analytical methods. Therefore, a reliable and efficient coupling scheme based on BEM was proposed to address this challenge, where, the Cartesian transformation method (CTM) has been implemented to calculate the domain integrals, and the generalized modified shift-splitting (GMSS) has been implemented for solving the linear systems arising from BEM. The calculation findings are depicted in graphical forms to display the impacts of temperature-dependent, anisotropy, piezoelectric, graded parameter and fractional parameter on the nonlinear photo thermal stress wave propagation in the considered structure. The numerical findings confirm the consistency and efficacy of the developed modeling methodology.

Fracture mechanics analysis of two-dimensional cracked thin structures (from micro- to nano-scales) by an efficient boundary element analysis

In this paper, the boundary element method (BEM) based on the elasticity theory is developed for fracture analysis of cracked thin structures with the relative thickness-to-length ratio in the micro- or nano-scales. A special crack-tip element technique is employed for the direct and accurate calculation of stress intensity factors (SIFs). The nearly singular integrals, which are crucial in applying the BEM for thin-structural problems, are calculated accurately by using a nonlinear coordinate transformation method. The present BEM procedure requires no remeshing procedure regardless of the thickness of thin structure. Promising SIFs results with only a small number of boundary elements can be achieved with the relative thickness of the thin film is as small as 10−9, which is sufficient for modeling most of the thin bodies as used in, for example, smart materials and micro/nano-electro-mechanical systems.

A new DBEM for solving crack problems in arbitrary dissimilar materials

In this paper, a new and efficient Dual Boundary Element Method (DBEM) named Dual Boundary-Interface Integral Equation Method (DBIIEM) is presented to solve crack problems in objects composed of arbitrary number of different materials. The conventional DBEM has been widely used in crack analysis and proved to be one of the most effective BEM for solving crack problems, however it cannot be directly employed to solve more complicated multi-medium crack problems. The presented DBIIEM can remedy this deficiency by dually collocating the displacement boundary-interface integral equation on one of the multi-medium crack surface and traction boundary-interface integral equation on the other. The newly derived displacement and traction boundary-interface integral equations are naturally fit for solving multi-medium elasticity problems, and independent each other, therefore they can be employed together in multi-medium crack analysis. The established DBIIEM inherits the advantage of single -region formulation of DBEM for solving more complicated crack problems. Complex stress intensity factors which characterize the oscillatory singularity of crack tips located at the interface crack are evaluated. Several numerical examples are given to verify correctness of the presented method.

DBEM computation of T-stress and mixed-mode SIFs using interaction integral technique

In this paper, dual boundary element method (DBEM) is combined with interaction integral (I-integral) to evaluate stress intensity factors (SIFs) and T-stress simultaneously for mixed-mode crack analyzing. DBEM is proved to be one of the most efficient boundary element methods (BEM) for solving crack problems with boundary-only discretization, however its current implementation is in lack of efficient methods to evaluate fracture parameters. The available geometrical extrapolation method and J-integral technique for calculating fracture parameters in combination with DBEM are imprecise and difficult to be implemented in mix-mode crack analysis. The presented combined DBEM and I-integral approach can be easily used to extract SIFs and T-stress for mixed-mode cracks, which does not need the decomposition of elastic field into symmetric and antisymmetric components. The fundamental basis lies in the introduction of auxiliary fields for some proper defined single mode crack, then fracture parameters are calculated by evaluating the formulated I-integral in terms of DBEM solution and auxiliary elastic fields. In this paper, the basis of DBEM is illustrated at first, then the explicit expressions for auxiliary fields of displacement gradients are derived, then an automatic scheme for generating the path of I-integral is presented. Several numerical examples are given to validate correctness of the presented method.

Profile reconstruction of a local defect in a groove structure and the theoretical limit under the vector diffraction theory.

The profile of a fine local defect in a periodic surface relief structure is reconstructed from a scattered wave. This defect cannot be imaged with an optical imaging system owing to the diffraction limit, and complicated multiscattering among the high-aspect-ratio grooves and the defect makes it difficult to reconstruct the profile using the scalar diffraction theory. We propose and numerically demonstrate a reconstruction algorithm by applying an efficient vector analysis method-the difference-field boundary element method. We also classify the profile according to the difficulty of reconstruction, which depends on the observation system and the noise level. Finally, this analysis provides the accuracy and limit of reconstruction under the vector diffraction theory.

Novel special crack-tip elements for interface crack analysis by an efficient boundary element method

This paper presents a new set of novel special crack-tip elements for analysis of interface cracks in linear elastic composite bimaterials by using the boundary element method (BEM). Such interface crack problems present some simulation difficulties based on the conventional numerical methods because the asymptotic crack-tip fields in this case exhibit an oscillatory behavior which is quite different from that in a cracked homogeneous material. The novel special crack-tip elements developed in this study contain the correct local asymptotic behavior of the displacement and stress fields at the tip of an interface crack, and can thus appropriately describe the oscillatory near-tip displacement and stress fields for cracked composite bimaterials. The novel special crack-tip elements are implemented into a BEM, which can significantly improve the computational accuracy of the complex stress intensity factor (SIF) for an interface crack even with a relatively coarse mesh. The proposed computational strategy for interface crack analysis based on the novel special crack-tip elements also enables a direct and accurate evaluation of the complex SIF at the collocation point extremely close to the crack-tip. Several numerical examples are presented and discussed to demonstrate that the present method based on the novel special crack-tip elements is pretty simple, highly accurate, and relatively robust with respect to the size of the used crack-tip elements.

Efficient isogeometric boundary element method for analysis of acoustic scattering from rigid bodies.

Boundary integral analysis of scattering from rigid bodies is well known. Analysis often proceeds along the following lines: representation of the geometry using a collection of triangles, representation of physics using low order ansatz functions defined on each triangle, and then solving the resulting discrete system. This prescription for the common solution stands out in terms of the low-order approximation of both geometry and representation of physics; specifically, both are C0. Taking inspiration from computer graphics literature, a framework wherein continuity of representation (both geometry and physics) can be as high as C2 is developed. In this paper, the steps necessary to develop such a iso-geometric (i.e., using the same basis functions for representing both geometry and physics) boundary integral solver are elucidated. In doing so, an efficient method based on a wideband fast multipole method to evaluate the required inner products and matrix vector products is proposed and demonstrated. Numerous examples are presented to highlight the benefits of the proposed approach.

Boundary element formulation of axisymmetric problems in vertically non-homogeneous solids subject to normal traction

This paper presents an efficient and accurate boundary element method (BEM) for the elastic analysis of axisymmetric problems in vertically non-homogeneous solids without or with cavity. This BEM uses the fundamental solution of the elastic field in a multilayered elastic solid induced by the body force uniformly concentrated at a circular ring. This solution is also called Yue's solution. The effective integration methods are used for dealing with the integrals in the discretized boundary integral equations. The discretization of the boundary surface uses one-dimensional boundary elements. It also adopts the infinite boundary element to take into account the influence of a far-field region on the boundary surface. Numerical verifications of displacements and stresses for three benchmark problems are conducted, which gives the excellent agreement with previously published results. Case studies are presented to numerically illustrate the influences of both vertically non-homogeneous elastic material properties and spherical cavity on the elastic fields induced by uniform tractions on the boundary surface. These numerical results show that this new BEM is a fast and simple numerical algorithm for accurately computing the axisymmetric elastic fields in vertically non-homogeneous solids with or without cavity induced by normal tractions.

A new convolution variational boundary element technique for design sensitivity analysis and topology optimization of anisotropic thermo-poroelastic structures

The main objective of this paper is to propose a new efficient time domain boundary element technique for design sensitivity analysis and topology optimization of anisotropic thermo-poroelastic structures. The spatial regularization scheme based on integration by parts is performed in Laplace domain in conjunction with time-domain convolution variational boundary element method (CVBEM). The generating optimization problem is solved using the Moving Asymptotes Method (MAM) with the adjoint variable method to obtain optimal material distribution, influence of anisotropy on the thermo-poroelastic stresses sensitivities. The results from the numerical model demonstrate the validity, efficiency and accuracy of the proposed technique.

A fast multi-layer boundary element method for direct numerical simulation of sound propagation in shallow water environments

Direct three-dimensional (3D) numerical simulations of acoustic fields in range-dependent shallow water environments remains a challenge due to environmental complexities and large computational cost. We develop an efficient 3D boundary element method (BEM) to solve the Helmholtz equation for shallow water acoustic propagation, which utilizes a Pre-corrected Fast Fourier Transform (PFFT) approach to reduce the computational effort from O(N2∼3) to O(Nlog⁡N) where N is the total number of boundary unknowns. To account for inhomogeneous media, the method allows for arbitrary number of coupled multi-layer BEM sub-domains. With O(Nlog⁡N) efficiency and the use of massively parallel high-performance computing platforms, we are able to conduct multi-layer 3D direct numerical simulations of low-mid frequency acoustics over kilometer ranges. We perform extensive validations of the method and provide two shallow water waveguide examples benchmarked against theoretical solutions. To illustrate the efficacy and usefulness of the PFFT-BEM method, we perform 3D large-scale direct numerical simulations to assess the performance of two established canonical models: axisymmetric coupled mode model for 3D seamount; and Kirchhoff approximation and perturbation theory for 3D rough surface scattering.