The Boundary Element Method for Ordinary State-Based Peridynamics

The peridynamics(PD),as a promising nonlocal continuum mechanics theory,shines in solving discontinuous problems.Up to now,various numerical methods,such as the peridynamic mesh-free particlemethod(PD-MPM),peridynamic finite element method(PD-FEM),and peridynamic boundary element method(PD-BEM),have been proposed.PD-BEM,in particular,outperforms other methods by eliminating spurious boundary softening,efficiently handling infinite problems,and ensuring high computational accuracy.However,the existing PD-BEM is constructed exclusively for bond-based peridynamics(BBPD)with fixed Poisson’s ratio,limiting its applicability to crack propagation problems and scenarios involving infinite or semi-infinite problems.In this paper,we address these limitations by introducing the boundary element method(BEM)for ordinary state-based peridynamics(OSPD-BEM).Additionally,we present a crack propagationmodel embeddedwithin the framework ofOSPD-BEM to simulate crack propagations.To validate the effectiveness of OSPD-BEM,we conduct four numerical examples:deformation under uniaxial loading,crack initiation in a double-notched specimen,wedge-splitting test,and threepoint bending test.The results demonstrate the accuracy and efficiency of OSPD-BEM,highlighting its capability to successfully eliminate spurious boundary softening phenomena under varying Poisson’s ratios.Moreover,OSPDBEMsignificantly reduces computational time and exhibits greater consistencywith experimental results compared to PD-MPM.

Enriched Constant Elements in the Boundary Element Method for Solving 2D Acoustic Problems at Higher Frequencies

The boundary element method(BEM)is a popular method for solving acoustic wave propagation problems,especially those in exterior domains,owing to its ease in handling radiation conditions at infinity.However,BEM models must meet the requirement of 6–10 elements per wavelength,using the conventional constant,linear,or quadratic elements.Therefore,a large storage size of memory and long solution time are often needed in solving higher-frequency problems.In this work,we propose two new types of enriched elements based on conventional constant boundary elements to improve the computational efficiency of the 2D acoustic BEM.The first one uses a plane wave expansion,which can be used to model scattering problems.The second one uses a special plane wave expansion,which can be used tomodel radiation problems.Five examples are investigated to showthe advantages of the enriched elements.Compared with the conventional constant elements,the new enriched elements can deliver results with the same accuracy and in less computational time.This improvement in the computational efficiency is more evident at higher frequencies(with the nondimensional wave numbers exceeding 100).The paper concludes with the potential of our proposed enriched elements and plans for their further improvement.

Generalized nth-Order Perturbation Method Based on Loop Subdivision Surface Boundary Element Method for Three-Dimensional Broadband Structural Acoustic Uncertainty Analysis

In this paper,a generalized nth-order perturbation method based on the isogeometric boundary element method is proposed for the uncertainty analysis of broadband structural acoustic scattering problems.The Burton-Miller method is employed to solve the problem of non-unique solutions that may be encountered in the external acoustic field,and the nth-order discretization formulation of the boundary integral equation is derived.In addition,the computation of loop subdivision surfaces and the subdivision rules are introduced.In order to confirm the effectiveness of the algorithm,the computed results are contrasted and analyzed with the results under Monte Carlo simulations(MCs)through several numerical examples.

Boundary Element Analysis forModeⅢCrack Problems of Thin-Walled Structures from Micro-to Nano-Scales

This paper develops a new numerical framework for modeⅢcrack problems of thin-walled structures by integrating multiple advanced techniques in the boundary element literature.The details of special crack-tip elements for displacement and stress are derived.An exponential transformation technique is introduced to accurately calculate the nearly singular integral,which is the key task of the boundary element simulation of thin-walled structures.Three numerical experiments with different types of cracks are provided to verify the performance of the present numerical framework.Numerical results demonstrate that the present scheme is valid for modeⅢcrack problems of thin-walled structures with the thickness-to-length ratio in the microscale,even nanoscale,regime.

Topology Optimization of Sound-Absorbing Materials for Two-Dimensional Acoustic Problems Using Isogeometric Boundary Element Method

In this work,an acoustic topology optimizationmethod for structural surface design covered by porous materials is proposed.The analysis of acoustic problems is performed using the isogeometric boundary elementmethod.Taking the element density of porousmaterials as the design variable,the volume of porousmaterials as the constraint,and the minimum sound pressure or maximum scattered sound power as the design goal,the topology optimization is carried out by solid isotropic material with penalization(SIMP)method.To get a limpid 0–1 distribution,a smoothing Heaviside-like function is proposed.To obtain the gradient value of the objective function,a sensitivity analysis method based on the adjoint variable method(AVM)is proposed.To find the optimal solution,the optimization problems are solved by the method of moving asymptotes(MMA)based on gradient information.Numerical examples verify the effectiveness of the proposed topology optimization method in the optimization process of two-dimensional acoustic problems.Furthermore,the optimal distribution of sound-absorbingmaterials is highly frequency-dependent and usually needs to be performed within a frequency band.

Error Analysis of A New Higher Order Boundary Element Method for A Uniform Flow Passing Cylinders

A higher order boundary element method(HOBEM)is presented for inviscid flow passing cylinders in bounded or unbounded domain.The traditional boundary integral equation is established with respect to the velocity potential and its normal derivative.In present work,a new integral equation is derived for the tangential velocity.The boundary is discretized into higher order elements to ensure the continuity of slope at the element nodes.The velocity potential is also expanded with higher order shape functions,in which the unknown coefficients involve the tangential velocity.The expansion then ensures the continuities of the velocity and the slope of the boundary at element nodes.Through extensive comparison of the results for the analytical solution of cylinders,it is shown that the present HOBEM is much more accurate than the conventional BEM.

Three-dimensional acoustic propagation model for shallow waters based on an indirect boundary element method

This work has a two-fold purpose.On the one hand,the theoretical formulation of a three-dimensional(3D)acoustic propagation model for shallow waters with a constant sound speed is presented,based on the boundary element method(BEM),which uses a half-space Green function instead of the more conventional free-space Green function.On the other hand,a numerical implementation is illustrated to explore the formulation in simple idealized cases,controlled by a few parameters,which provides necessary tests for the accuracy and performance of the model.The half-space Green's function,which has been previously used in scattering and diffraction,adds terms to the usual expressions of the integral operators without altering their continuity properties.Verifications against the wavenumber integration solution of the Pekeris waveguide suggest that the model allows an adequate prediction for the acoustic field.Likewise,numerical experiments in relation to the necessary mesh size for the description of the water-marine sediment interface lead to the conclusion that a transmission loss prediction with acceptable accuracy can be obtained with the use of a limited mesh around the desired evaluation region.

BOUNDARY ELEMENT ANALYSIS OF CONTACT PROBLEMS USING ARTIFICIAL BOUNDARY NODE APPROACH

An improved version of the regular boundary element method, the artificial boundary node approach, is derived. A simple contact algorithm is designed and implemented into the direct boundary element, regular boundary element and artificial boundary node approaches. The exisiting and derived approaches are tested using some case studies. The results of the artificial boundary node approach are compared with those of the existing boundary element program, the regular element approach, ANSYS and analytical solution whenever possible. The results show the effectiveness of the artificial boundary node approach for a wider range of boundary offsets.

A DOMAIN DECOMPOSITION ALGORITHM WITH FINITE ELEMENT-BOUNDARY ELEMENT COUPLING

A domain decomposition algorithm coupling the finite element and the boundary element was presented. It essentially involves subdivision of the analyzed domain into sub-regions being independently modeled by two methods, i.e., the finite element method （FEM） and the boundary element method （BEM）. The original problem was restored with continuity and equilibrium conditions being satisfied on the interface of the two sub-regions using an iterative algorithm. To speed up the convergence rate of the iterative algorithm, a dynamically changing relaxation parameter during iteration was introduced. An advantage of the proposed algorithm is that the locations of the nodes on the interface of the two sub-domains can be inconsistent. The validity of the algorithm is demonstrated by the consistence of the results of a numerical example obtained by the proposed method and those by the FEM, the BEM and a present finite element-boundary element （FE-BE） coupling method.

Improved design of special boundary elements for T-shaped reinforced concrete walls

This study examines the design provisions of the Chinese GB 50011-2010 code for seismic design of buildings for the special boundary elements of T-shaped reinforced concrete walls and proposes an improved design method. Comparison of the design provisions of the GB 50011-2010 code and those of the American code ACI 318-14 indicates a possible deficiency in the T-shaped wall design provisions in GB 50011-2010. A case study of a typical T-shaped wall designed in accordance with GB 50011-2010 also indicates the insufficient extent of the boundary element at the non-flange end and overly conservative design of the flange end boundary element. Improved designs for special boundary elements ofT-shaped walls are developed using a displacement-based method. The proposed design formulas produce a longer boundary element at the non-flange end and a shorter boundary element at the flange end, relative to those of the GB 50011-2010 provisions. Extensive numerical analysis indicates that T-shaped walls designed using the proposed formulas develop inelastic drift of 0.01 for both cases of the flange in compression and in tension.

Forward Calculation of 2-D and 3-D Structures with a Cover by the Boundary Element Method while Using Electrical Methods

This paper develops the boundary element method, the authors employ two-layered earth Green 's functions as the weighting functions of residual and derive boundary integral equations. The forward problems of point sources on 2 - D and 3-D structures with an influencing cover are solved by this method. The results show that this method markedly improves the original boundary element method. The features of the improved method are greater numerical accuracy and much smaller systems of equations and thus considerable savings for the storage capacity of computers, allowing us to solve the above problems with only ordinary microcomputers. The results in this paper extend the scope of applying the boundary element method while using electrical methods for geophysical prospecting.

Three-Dimensional Boundary Element Method Applied to Nonlinear Wave Transformation

For higher accuracy in simulating the transformation of three dimensional waves, in consideration of the advantages of constant panels and linear elements, a combined boundary elements is applied in this research. The method can be used to remove the transverse vibration due to the accumulation of computational errors. A combined boundary condition of sponge layer and Sommerfeld radiation condition is used to remove the reflected waves from the computing domain. By following the water particle on the water surface, the third order Stokes wave transform is simulated by the numerical wave flume technique. The computed results are in good agreement with theoretical ones.

Simulation of the seismic response of sedimentary basins with constant-gradient velocity along arbitrary direction using boundary element method:SH case

We presented a boundary element method using the approximate analytical Green＇s function given by Sanchez-Sesma et al. Coordinate transform is introduced to extend the method to deal with the model with constant-gradient velocity along oblique direction. The method is validated by comparing the numerical results with other independent methods. This method provides a useful tool for analyzing local site effects. We computed seismic response for two series of models. The results in both frequency and time domains are analyzed and show complex amplification patterns. The fundamental mode of resonance is dependent not only on the velocity at the free surface but also on the velocity distribution of the whole basin. For the higher modes of vibration the heterogeneous basin also has its own characteristic.

BOUNDARY ELEMENT METHOD FOR MOVING AND ROLLING CONTACT OF 2D ELASTIC BODIES WITH DEFECTS

A scheme of boundary element method for moving contact of two-dimensional elastic bodies using conforming discretization is presented. Both the displacement and the traction boundary conditions are satisfied on the contacting region in the sense of discretization. An algorithm to deal with the moving of the contact boundary on a larger possible contact region is presented. The algorithm is generalized to rolling contact problem as well. Some numerical examples of moving and rolling contact of 2D elastic bodies with or without friction, including the bodies with a hole-type defect, are given to show the effectiveness and the accuracy of the presented schemes.

Isogeometric Boundary Element Analysis for 2D Transient Heat Conduction Problem with Radial Integration Method

This paper presents an isogeometric boundary element method(IGABEM)for transient heat conduction analysis.The Non-Uniform Rational B-spline(NURBS)basis functions,which are used to construct the geometry of the structures,are employed to discretize the physical unknowns in the boundary integral formulations of the governing equations.Bezier extraction technique is employed to accelerate the evaluation of NURBS basis functions.We adopt a radial integration method to address the additional domain integrals.The numerical examples demonstrate the advantage of IGABEM in dimension reduction and the seamless connection between CAD and numerical analysis.

Monte Carlo Simulation of Fractures Using Isogeometric Boundary Element Methods Based on POD-RBF

This paper presents a novel framework for stochastic analysis of linear elastic fracture problems.Monte Carlo simulation(MCs)is adopted to address the multi-dimensional uncertainties,whose computation cost is reduced by combination of Proper Orthogonal Decomposition(POD)and the Radial Basis Function(RBF).In order to avoid re-meshing and retain the geometric exactness,isogeometric boundary element method(IGABEM)is employed for simulation,in which the Non-Uniform Rational B-splines(NURBS)are employed for representing the crack surfaces and discretizing dual boundary integral equations.The stress intensity factors(SIFs)are extracted by M integral method.The numerical examples simulate several cracked structures with various uncertain parameters such as load effects,materials,geometric dimensions,and the results are verified by comparison with the analytical solutions.

Introducing an effective coherence function to generate non-uniform ground motion on topographic site using time-domain boundary element method

In this study,a comprehensive parametric analysis was performed on non-uniform excitation of V-shaped topography using the boundary element method in time domain.For this purpose,wave scattering analysis was carried out on a topography subjected to the SV-wave for different predominant frequencies and shape ratios.Based on the numerical results,new coherence and time delay functions are proposed to generate non-uniform ground motion for topographic irregularities.The efficiency and accuracy of the proposed functions for real engineering problems are indicated by comparison with observations reported in previous literature.

Machine Learning Enhanced Boundary Element Method:Prediction of Gaussian Quadrature Points

This paper applies a machine learning technique to find a general and efficient numerical integration scheme for boundary element methods.A model based on the neural network multi-classification algorithmis constructed to find the minimum number of Gaussian quadrature points satisfying the given accuracy.The constructed model is trained by using a large amount of data calculated in the traditional boundary element method and the optimal network architecture is selected.The two-dimensional potential problem of a circular structure is tested and analyzed based on the determined model,and the accuracy of the model is about 90%.Finally,by incorporating the predicted Gaussian quadrature points into the boundary element analysis,we find that the numerical solution and the analytical solution are in good agreement,which verifies the robustness of the proposed method.

A Combined Shape and Topology Optimization Based on Isogeometric Boundary Element Method for 3D Acoustics

A combined shape and topology optimization algorithm based on isogeometric boundary element method for 3D acoustics is developed in this study.The key treatment involves using adjoint variable method in shape sensitivity analysis with respect to non-uniform rational basis splines control points,and in topology sensitivity analysis with respect to the artificial densities of sound absorption material.OpenMP tool in Fortran code is adopted to improve the efficiency of analysis.To consider the features and efficiencies of the two types of optimization methods,this study adopts a combined iteration scheme for the optimization process to investigate the simultaneous change of geometry shape and distribution of material to achieve better noise control.Numerical examples,such as sound barrier,simple tank,and BeTSSi submarine,are performed to validate the advantage of combined optimization in noise reduction,and to demonstrate the potential of the proposed method for engineering problems.

A boundary element method for the simulation of non-spherical bubbles and their interactions near a free surface

The basic principle and numerical technique for simulating two three-dimensional bubbles near a free surface are studied in detail by using boundary element method. The singularities of influence coefficient matrix are eliminated using coordinate transformation and so-called 4π rule. The solid angle for the open surface is treated in direct method based on its definition. Several kinds of configurations for the bubbles and free surface have been investigated. The pressure contours during the evolution of bubbles are obtained in our model and can better illuminate the mechanism underlying the motions of bubbles and free surface. The bubble dynamics and their interactions have close relation with the standoff distances, buoyancy parameters and initial sizes of bubbles. Completely different bubble shapes, free surface motions, jetting patterns and pressure distributions under different parameters can be observed in our model, as demon- strated in our calculation results.