Regularized Boundary Research Articles

A novel high‐order collocation indirect boundary element method based on the Leis formulation for three‐dimensional high frequency exterior acoustic problems

AbstractHigh‐order finite element method (FEM) has recently achieved significant success in solving acoustic problems and has proved to be accurate, robust, and capable of reducing the pollution effect. Inspired by the high‐order FEM, this study presents a novel high‐order collocation indirect BEM for solving three‐dimensional high‐frequency exterior acoustic problems. Due to the difficulty of evaluating hypersingular integrals on curved surface elements with complex density functions, the high‐order BEM scheme for acoustic problems has long been a challenging task, especially for relatively large‐scale problems. In the developed scheme, the Leis formulation is used to overcome the well‐known nonuniqueness problem in the indirect BEM, and an auxiliary integral equation method is presented and employed to regularize the hypersingular integrals in this formulation. The derived regularized boundary integral formulation is mathematically simple and amenable to the elements of arbitrary shape and arbitrary variable interpolation order, and leads to a simple, efficient, and robust algorithm for the calculation of all integrals. Several benchmark examples are investigated to verify the performance of the proposed method. Results demonstrate that the present method is very effective in stimulating the far‐field solution with large wavenumber.

РАСЧЕТЫ ДЕЙСТВИЯ СИЛЫ НА ПОРОУПРУГОЕ ТРЕХФАЗНОЕ ПОЛУПРОСТРАНСТВО

The mathematical formulation of the boundary value problem in the images of the linear three-dimensional isotropic dynamic theory of poroelasticity is given when the material of the medium is partially saturated. The poroelastic medium is represented by a model of a heterogeneous material consisting of a phase of an elastic matrix and two phases of fillers – liquid and gas filling the pore system. A method for the numerical inversion of the Laplace transform is described. The description of the boundary-element approach is given. A regularized boundary integral equation is given. A discrete analogue of the boundary integral equation is written down. The discrete analog is parametrized by the complex variable of the Laplace integral transform. As an example, the problem of the action of a vertical force uniformly distributed on a square area on a partially saturated poroelastic half-space is considered. The choice of the computational boundary-element model made it possible to obtain verified numerical results. The effect of saturation on the amplitudes of the nonstationary wave response and on the velocities of the response waves is noted. The behavior of the wave response with an isotropic, homogeneous, poroelastic half-space moving away from the loading source is demonstrated. The problem of the action of a vertical force on a partially saturated poroelastic half-space weakened by a cubic cavity is considered. The influence of half-space attenuation on the wave surface response is analyzed. There is an analysis of the influence of depth on the dynamic wave response, as well as the influence of the phase of the poroelastic material.

An isogeometric finite element-boundary element approach for the vibration analysis of submerged thin-walled structures

In this paper, the isogeometric formulations of the finite element and boundary element methods are applied to the dynamic analysis of thin-walled structures submerged in an infinite, inviscid, and incompressible fluid medium. This fluid–structure interaction problem is decoupled using the modal analysis technique, and the fluid effect on the structure is taken into account through the generalized added mass matrix. The structure is modeled with NURBS-based Kirchhoff–Love shell elements. The fluid response is computed using a regularized boundary integral equation. We take advantage of the geometry preserving property of the NURBS refinement techniques to reduce the computational cost without the need for a projection scheme. The implementation is benchmarked with three test cases, and good accuracy is obtained for a relatively low number of degrees of freedom.

Sloshing force in a rectangular tank with porous media

The porous media in a water tank will affect the dynamic characteristics of the sloshing behavior significantly. The sloshing force is one of the most important features, especially when resonance occurs. In this paper, the analytical equivalent mechanical model and numerical regularized boundary integral method (RBIM) are presented and verified by a resonant test. The experimental measurements show the analytical solution is useful for the linear-flow regime, while the numerical method provides accurate prediction for the nonlinear-flow case. The quadratic equation for the pressure drop due to porous media can be available as the suitable parameters are determined.

Spectrum of end of the world branes in holographic BCFTs

We study overlaps between two regularized boundary states in conformal field theories. Regularized boundary states are dual to end of the world branes in an AdS black hole via the AdS/BCFT. Thus they can be regarded as microstates of a single sided black hole. Owing to the open-closed duality, such an overlap between two different regularized boundary states is exponentially suppressed as leftlangle left.{psi}_aright|{psi}_brightrangle sim {e}^{-Oleft({h}_{ab}^{left(min right)}right)} , where {h}_{ab}^{left(min right)} is the lowest energy of open strings which connect two different boundaries a and b. Our gravity dual analysis leads to {h}_{ab}^{left(min right)} = c/24 for a pure AdS3 gravity. This shows that a holographic boundary state is a random vector among all left-right symmetric states, whose number is given by a square root of the number of all black hole microstates. We also perform a similar computation in higher dimensions, and find that {h}_{ab}^{left(min right)} depends on the tensions of the branes. In our analysis of holographic boundary states, the off diagonal elements of the inner products can be computed directly from on-shell gravity actions, as opposed to earlier calculations of inner products of microstates in two dimensional gravity.

Generating adversarial examples with elastic-net regularized boundary equilibrium generative adversarial network

To improve the attack success rate and image perceptual quality of adversarial examples against deep neural networks(DNNs), we propose a new Generative Adversarial Network (GAN) based attacker, named Elastic-net Regularized Boundary Equilibrium Generative Adversarial Network(ERBEGAN). Recent studies have shown that DNNs are easy to attack by adversarial examples(AEs) where benign images with small-magnitude perturbations mislead DNNs to incorrect results. A number of methods are proposed to generate AEs, but how to generate them with high attack success rate and perceptual quality needs more effort. Most attackers generate AEs by restricting L2-norm and L∞-norm of adversarial perturbations. However, very few works have been developed on L1 distortion matrix which encourages sparsity in the perturbation. In this paper, we penalize both L2-norm and L1-norm of perturbation as Elastic-Net regularization to improve the diversity and robustness of AEs. We further improve GAN by minimizing the additional pixel-wise loss derived from the Wasserstein distance between benign and adversarial auto-encoder loss distributions. Extensive experiments and visualizations on several datasets show that the proposed ERBEGAN can yield higher attack success rates than the state-of-the-art GAN-based attacker AdvGAN under the semi-whitebox and black-box attack settings. Besides, our method efficiently generates diverse adversarial examples that are more perceptually realistic.

An improved minimum bounding rectangle algorithm for regularized building boundary extraction from aerial LiDAR point clouds with partial occlusions

ABSTRACT Airborne Light Detection And Ranging (LiDAR) point cloud is an important data source for building 3D digital cities and can be used to acquire and update urban buildings, roads and etc. However, it is difficult to extract complete and accurate building boundaries from airborne LiDAR point clouds due to partial occlusion, which is mainly caused by adjacent tall trees, particularly in spring and summer. In this paper, we propose an improved minimum bounding rectangle (IMBR) algorithm to extract complete and accurate regularized building boundaries with and without partial occlusion from aerial LiDAR point clouds. The new algorithm only uses LiDAR point cloud and doesn’t need any additional data source. In addition, the algorithm can be applied to buildings with complex shapes. To test the proposed algorithm and compare it with the recursive minimum bounding rectangle (RMBR) algorithm, three datasets with different types of partial occlusions and different shapes were tested. The experimental results show that IMBR can successfully extract the complete and accurate regularized building boundary with or without partial occlusion, and its accuracy is equal to that of RMBR algorithm.

The effect of boundary conditions on the accuracy and stability of the numerical solution of fluid flows by Lattice-Boltzmann method

The aim of this study is to investigate the effect of boundary conditions on the accuracy and stability of the numerical solution of fluid flows in the context of single relaxation time Lattice Boltzmann method (SRT-LBM). The fluid flows are simulated using regularized, no-slip, Zou-He and bounce back boundary conditions for straight surfaces in a lid driven cavity and the two-dimensional flow in a channel. The solutions for all types of the boundary conditions show good agreement with numerical references and exact solutions. The cavity pressure contours at low relaxation time show drastic perturbations for Zou-He boundary condition, whereas, the perturbation is ignorable for regularized boundary condition. At High Reynolds number, severe velocity gradients are major reason for numerical instabilities. Therefore, regularized boundary condition, which considers the velocity gradient in its calculation, has better numerical stability comparing the Zou-He boundary condition. Overall, the selection of appropriate boundary condition depends on the flow regime and Geometry. The proper boundary conditions at low Reynolds numbers are Zou-He and bounce back boundary conditions, and at high Reynolds numbers, regularized and no-slip boundary conditions are recommended.

Nonlinear Dynamic Characteristics of Rectangular and Cylindrical TLDs

AbstractThe nonlinear dynamic characteristics of rectangular and cylindrical tuned liquid dampers (TLDs) were studied by the so-called regularized boundary integral method (RBIM). An artificial dam...

A regularized boundary element formulation with weighted-collocation and higher-order projection for 3D time-domain elastodynamics

To advance Time-Domain Boundary Element Methods (TD-BEMs), a generalized direct time-integration solution method for three-dimensional elastodynamics is presented in this paper. On the basis of a general decomposition of time-dependent point-load Green's functions into a singular and regular part, a regularized boundary integral equation for the time domain is formulated and implemented via a variable-weight multi-step collocation scheme that allows for different orders of time projection for the boundary displacements and tractions. The benefits and possibilities of improved performance by suitable collocation weights and the solution projection choices are illustrated via two benchmark finite-domain and infinite-domain problems.

Development of boundary-element time-step scheme in solving 3D poroelastodynamics problems

The development of time-step boundary-element scheme for the three dimensional boundaryvalue problems of poroelastodynamics is presented. The poroelastic continuum is described using Biot’s mathematical model. Poroelastic material is assumed to consist of a solid phase constituting an elastic formdefining skeleton and carrying most of the loading, and two fluid phases filling the pores. Dynamic equations of the poroelastic medium are written for unknown functions of displacement of the elastic skeleton and pore pressures of the filling materials. Green’s matrices and, based on it, boundary integral equations are written in Laplace domain. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. Boundary element scheme is based on time-step method of numerical inversion of Laplace transform. A modification of the time-step scheme on the nodes of Runge-Kutta methods is considered. The Runge-Kutta scheme is exemplified with 2-and 3-stage Radau schemes. The results of comparing the two schemes in analyzing a numerical example are presented.

A meshless average source boundary node method for steady-state heat conduction in general anisotropic media

The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.

Evaluation of hypersingular and nearly singular integrals in the Isogeometric Boundary Element Method for acoustics

Special integration routines for hypersingular and nearly singular integrals that arise in an Isogeometric Boundary Element Method (IGABEM) are presented. The IGABEM is applied to an acoustic problem in the frequency domain that corresponds to the Helmholtz equation in three dimensions. A major topic of such an acoustical BEM is the non-uniqueness of the conventional boundary integral equation, which can be corrected by the Burton–Miller formulation at the cost of a hypersingular integral that requires the presented routines. The combination of the BEM with an Isogeometric Analysis (IGA) is advantageous, since the exact CAD description is directly incorporated to the numerical simulation. As the geometry is represented exactly, the integration routines become even more relevant and cannot be masked by the geometrical approximation. For the conventional approximation with plane Lagrange elements, a variety of different integration routines exist, but are not investigated in the scope of an IGA. Therefore, the hypersingular integration scheme of Guiggiani is adapted to be applicable to this new methodology. A direct evaluation of the hypersingular integral is possible, instead of an integration over the complete surface, as in the often used regularized boundary integral equation. Additionally, the sinh-transformation for nearly singular integrals is introduced to achieve more accurate results with the same number of integration points as the conventional Gaussian integration. The new formulation is analyzed under the special case of the oscillatory kernels in acoustics.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

Boundary stress analysis using a new regularized boundary integral equation for three-dimensional elasticity problems

This paper presents new regularized boundary integral equations (BIEs) for elastic displacement gradients in three dimensions and then combines them by the generalized Hooke’s law to calculate the boundary stress. In the new regularized BIEs, two special tangential vectors are designed with the normal vector to construct a transformation system. Based on this system, the displacement gradient in any direction can be transformed into a linear combination of the normal gradient and tangential gradients along the two special vectors. Moreover, a theorem related to some integral properties of the fundamental solution is introduced. Finally, the regularized indirect BIEs are developed by using the above-mentioned technique of linear combination and theorem. The proposed method has some advantages over the direct boundary element method, such as the relaxed continuity requirement of density function, no hypersingular integral, and being available to calculate the displacement gradient in any direction. The numerical implementation of the developed integral equation is provided, and the accuracy and convergence of the approach are also illustrated through four numerical examples.

Solvability of the regularized steady problem of the spatial motions of multicomponent viscous compressible fluids

We consider the boundary value problem arising in the analysis of steady barotropic motions of a viscous compressible multifluid in a bounded domain of the three-dimensional Euclidean space. We establish the existence of strong solutions to the regularized boundary value problem.

Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity

SummaryThe present work addresses shape sensitivity analysis and optimization in two‐dimensional elasticity with a regularized isogeometric boundary element method (IGABEM). Non‐uniform rational B‐splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations. With the advantage of tight integration of design and analysis, the application of IGABEM in shape optimization reduces the mesh generation/regeneration burden greatly. The work is distinct from the previous literatures in IGABEM shape optimization mainly in two aspects: (1) the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations, eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives, which were the main implementation difficulty in IGABEM, and (2) although based on the same Computer Aided Design (CAD) model, the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh, thus achieving a balance between less design variables for efficiency and refined mesh for accuracy. This technique was initially used in isogeometric finite element method and was incorporated into the present IGABEM implementation. Copyright © 2016 John Wiley & Sons, Ltd.

A new regularized boundary integral equation for three-dimensional potential gradient field

This paper presents a new regularized boundary integral equation (BIE) method for three-dimensional (3D) potential gradient field. For this method, we firstly construct two special tangential vectors, and then provide a characteristics theorem with respect to the contour integrations of normal and tangential gradients of the fundamental solution. Finally, a new regularized boundary integral equation with indirect unknowns is derived by using the characteristics theorem and a limit theorem. Compared with the direct boundary element method (BEM), the proposed method has three new features: (1) the continuity requirements of density functions are reduced from C1, α to C0, α; (2) the BIE does not involve the hypersingular (HFP) integral and thus its numerical evaluation is more easy and precise; (3) any potential gradients on the boundary, not limited to normal gradients, can now be calculated. Numerical results illustrate that the present method is computationally efficient, accurate, and convergent with an increasing number of boundary elements.

Computing the Casimir force using regularized boundary integral equations

In this paper we use a novel regularization procedure to reduce the calculation of the Casimir force for 2D scalar fields between compact objects to the solution of a classical integral equation defined on the boundaries of the objects. The scalar fields are subject to Dirichlet boundary conditions on the object boundaries. We test the integral equation by comparing with what we get for parallel plates, concentric circles and adjacent circles using mode summation and the functional integral method. We show how symmetries in the shapes and configuration of boundaries can easily be incorporated into our method and that it leads to fast evaluation of the Casimir force for symmetric situations.

A new method for numerical evaluation of nearly singular integrals over high-order geometry elements in 3D BEM

This work presents a new method for numerical computation of the two-dimensional nearly singular integrals by using the eight-node second-order quadrilateral surface elements in 3D BEM. A new indirect regularized boundary element formulation excluding the CPV (Cauchy Principal Value) and HFP (Hadamard-Finite-Part) integrals is proposed. Based on this, a new approximation formula of the distance from the fixed calculation point to a generic point of the aforementioned surface geometry elements is developed firstly, and then the exponential transformation, which has been widely employed in 2D BEM, is extended to 3D BEM to remove the near singularities of integrands for considered integrals. Several numerical examples are given to verify the high efficiency and the stability of the proposed scheme.