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.

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.

Resolving Domain Integral Issues in Isogeometric Boundary Element Methods via Radial Integration:A Study of Thermoelastic Analysis

The paper applied the isogeometric boundary element method(IGABEM)to thermoelastic problems.The Non-Uniform Rational B-splines(NURBS)used to construct geometric models are employed to discretize the boundary integral formulation of the governing equation.Due to the existence of thermal stress,the domain integral term appears in the boundary integral equation.We resolve this problem by incorporating radial integration method into IGABEM which converts the domain integral to the boundary integral.In this way,IGABEM can maintain its advantages in dimensionality reduction and more importantly,seamless integration of CAD and numerical analysis based on boundary representation.The algorithm is verified by numerical examples.

Convergence Properties of Local Defect Correction Algorithm for the Boundary Element Method

Sometimes boundary value problems have isolated regions where the solution changes rapidly.Therefore,when solving numerically,one needs a fine grid to capture the high activity.The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid.One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique.The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid.The algorithm is relatively new and its convergence properties have not been studied for the boundary element method.In this paper the objective is to determine convergence properties of the algorithm for the boundary element method.First,we formulate the algorithm as a fixed point iterative scheme,which has also not been done before for the boundary element method,and then study the properties of the iteration matrix.Results show that we can always expect convergence.Therefore,the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.

Subdivision Surface-Based Isogeometric Boundary Element Method for Steady Heat Conduction Problems with Variable Coefficient

The present work couples isogeometric analysis(IGA)and boundary element methods(BEM)for three dimensional steady heat conduction problems with variable coefficients.The Computer-Aided Design(CAD)geometries are built by subdivision surfaces,and meantime the basis functions of subdivision surfaces are employed to discretize the boundary integral equations for heat conduction analysis.Moreover,the radial integration method is adopted to transform the additional domain integrals caused by variable coefficients to the boundary integrals.Several numerical examples are provided to demonstrate the correctness and advantages of the proposed algorithm in the integration of CAD and numerical analysis.

SINGULAR INTEGRAL EQUATIONS AND BOUNDARY ELEMENT METHOD OF CRACKS IN THERMALLY STRESSED PLANAR SOLIDS

Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, and then,its numerical method is proposed by the numerical method of the singular integral equations combined with boundary element method. Moreover, the singular nature of temperature gradient field near the crack front is proved by the main-part analysis method of the singular integral equation, and the singular temperature gradients are exactly obtained. Finally, several typical examples calculated.

CALCULATION FOR PATH-DOMAIN INDEPENDENT J INTEGRAL WITH ELASTO-VISCOPLASTIC CONSISTENT TANGENT OPERATOR CONCEPT-BASED BOUNDARY ELEMENT METHODS

This paper presents an elasto-viscoplastic consistent tangent operator (CTO) based boundary element formulation, and application for calculation of path-domain independentJ integrals (extension of the classicalJ integrals) in nonlinear crack analysis. When viscoplastic deformation happens, the effective stresses around the crack tip in the nonlinear region is allowed to exceed the loading surface, and the pure plastic theory is not suitable for this situation. The concept of consistency employed in the solution of increment viscoplastic problem, plays a crucial role in preserving the quadratic rate asymptotic convergence of iteractive schemes based on Newton's method. Therefore, this paper investigates the viscoplastic crack problem, and presents an implicit viscoplastic algorithm using the CTO concept in a boundary element framework for path-domain independentJ integrals. Applications are presented with two numerical examples for viscoplastic crack problems andJ integrals.

REGULARIZATION OF NEARLY SINGULAR INTEGRALS IN THE BOUNDARY ELEMENT METHOD OF POTENTIAL PROBLEMS

A general algorithm is applied to the regularization of nearly singular integrals in the boundary element method of planar potential problems. For linear elements, the strongly singular and hypersingular integrals of the interior points very close to boundary were categorized into two forms. The factor leading to the singularity was transformed out of the integral representations with integration by parts, so non-singular regularized formulas were presented for the two forms of integrals. Furthermore, quadratic elements are used in addition to linear ones. The quadratic element very close to the internal point can be divided into two linear ones, so that the algorithm is still valid. Numerical examples demonstrate the effectiveness and accuracy of this algorithm. Especially for problems with curved boundaries, the combination of quadratic elements and linear elements can give more accurate results.

NONSINGULAR KERNEL BOUNDARY ELEMENT METHOD FOR THIN-PLATE BENDING PROBLEMS

In this paper, the nonsingular fundamental solutions were obtained from Fourier series under some given conditions. These solutions can be taken as the kernels of integral equation. So a new boundary element method was presented, with which all kinds of thin-plate bending problems can be solved, even with complicated loadings and sinuous boundaries. The calculation is much simpler and more accurate.

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

The isogeometric boundary element technique(IGABEM)is presented in this study for steady-state inhomogeneous heat conduction analysis.The physical unknowns in the boundary integral formulations of the governing equations are discretized using non-uniform rational B-spline(NURBS)basis functions,which are utilized to build the geometry of the structures.To speed up the assessment of NURBS basis functions,the Bezier extraction´approach is used.To solve the extra domain integrals,we use a radial integration approach.The numerical examples show the potential of IGABEM for dimension reduction and smooth integration of CAD and numerical analysis.

Numerical Aspects of Isogeometric Boundary Element Methods:(Nearly)Singular Quadrature,Trimmed NURBS and Surface Crack Modeling

This work presents some numerical aspects of isogeometric boundary element methods(IGABEM).The behavior of hyper-singular and nearly-singular integration is first explored on the distorted NURBS surface.Several numerical treatments are proposed to enhance the quadrature in the framework of isogeometric analysis.Then a numerical implementation of IGABEM on the trimmed NURBS is detailed.Based on this idea,the surface crack problem is modeled incorporation with the phantom element method.The proposed method allows the crack to intersect with the boundary of the body while preserving the original parametrization of the NURBS-based CAD geometry.

Radial integral boundary element method for simulating phase change problem with mushy zone

A radial integral boundary element method(BEM)is used to simulate the phase change problem with a mushy zone in this paper.Three phases,including the solid phase,the liquid phase,and the mushy zone,are considered in the phase change problem.First,according to the continuity conditions of temperature and its gradient on the liquid-mushy interface,the mushy zone and the liquid phase in the simulation can be considered as a whole part,namely,the non-solid phase,and the change of latent heat is approximated by heat source which is dependent on temperature.Then,the precise integration BEM is used to obtain the differential equations in the solid phase zone and the non-solid phase zone,respectively.Moreover,an iterative predictor-corrector precise integration method(PIM)is needed to solve the differential equations and obtain the temperature field and the heat flux on the boundary.According to an energy balance equation and the velocity of the interface between the solid phase and the mushy zone,the front-tracking method is used to track the move of the interface.The interface between the liquid phase and the mushy zone is obtained by interpolation of the temperature field.Finally,four numerical examples are provided to assess the performance of the proposed numerical method.

Quadrature Rules for Functions with a Mid-Point Logarithmic Singularity in the Boundary Element Method Based on the <i>x = t^p</i>Substitution

Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = tp, where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high value for p typically gives more accurate results.

Element nodal computation-based radial integration BEM for non-homogeneous problems

This paper presents a new strategy of using the radial integration boundary element method （RIBEM） to solve non-homogeneous heat conduction and thermoelasticity problems. In the method, the evaluation of the radial in-tegral which is used to transform domain integrals to equivalent boundary integrals is carried out on the basis of elemental nodes. As a result, the computational time spent in evaluating domain integrals can be saved considerably in comparison with the conventional RIBEM. Three numerical examples are given to demonstrate the correctness and computational efficiency of the proposed approach.

Geometric Conversion Approach for the Numerical Evaluation of Hypersin gular and Nearly Hypersingular Boundary Integrals over Curved Surface Boundary Elements

With the aid of the properties of the hypersingular kernels, a geometric conversion approach was presented in this paper. The conversion leads to a general approach for the accurate and reliable numerical evaluation of the hypersingular surface boundary integrals encountered in a variety of applications with boundary element method. Based on the conversion, the hypersingularity in the boundary integrals could be lowered by one order, resulting in the simplification of the computer code. Moreover, an integral transformation was introduced to damp out the nearly singular behavior of the kernels by the distance function defined in the local polar coordinate system for the nearly hypersingular case. The approach is simple to use, which can be inserted readily to computer code, thus getting rid of the dull routine deduction of formulae before the numerical implementations, as the expressions of these kernels are in general complicated. The numerical examples were given in three dimensional elasticity, verifying the effectiveness of the proposed approach, which makes it possible to observe numerically the behavior of the boundary integral values with hypersingular kernels across the boundary.

A New Iterative Method for Multi-Moving Boundary Problems Based Boundary Integral Method

The present paper deals with very important practical problems of wide range of applications. The main target of the present paper is to track all moving boundaries that appear throughout the whole process when dealing with multi-moving boundary problems continuously with time up to the end of the process with high accuracy and minimum number of iterations. A new numerical iterative scheme based the boundary integral equation method is developed to track the moving boundaries as well as compute all unknowns in the problem. Three practical applications, one for vaporization and two for ablation were solved and their results were compared with finite element, heat balance integral and the source and sink results and a good agreement were obtained.

基于特征分区的奇异域积分单元细分法

针对传统方法难以解决积分方程中的奇异性问题,提出一种基于特征分区的奇异域积分单元细分法,该方法基于体二叉树数据结构对不同类型体单元自适应细分,能精确计算任意源点位置的三维奇异积分,消除积分的奇异性.在笛卡尔坐标系下,通过在源点构建包围盒对体单元特征分区,将体单元划分为腔面投影区域和单元细分区域,依照细分准则对单元细分区域递归细分,采用腔面重构算法和投影算法,重新在源点附近生成高质量的积分子单元.数值算例表明,该方法的积分计算精度、稳定性优于传统单元细分方法.

耦合边界元法和等效源法的稳健CHIEF法

针对声学边界元法中解的非唯一性和奇异积分问题,基于组合亥姆霍兹积分方程公式(combined helmholtz integral equation formulation,CHIEF)法思想,将常规边界元方程和等效源方程进行联立,并利用两者方程系数矩阵间的耦合等价关系,间接替换计算常规边界元法中的奇异系数矩阵,进而提出一种具有全频域唯一解、高计算精度和高稳定性的耦合CHIEF法。该方法将等效源方程作为补充方程,不仅解决了传统CHIEF法内点补充方程失效的问题,而且矩阵的间接替换计算避免了直接计算奇异积分,显著提高了计算效率和精度。通过声辐射和声散射的典型算例对比了所提方法、常规边界元法、常规Burton-Miller法和等效源法的计算效果。结果表明,所提方法不仅在全波数域内均能获得唯一解,且其计算精度和效率均优于常规边界元法和常规Burton-Miller方法,其系数矩阵条件数远低于等效源法。

A Non-Singular Boundary Element Method for Interactions between Acoustical Field Sources and Structures

Localized point sources(monopoles)in an acoustical domain are implemented to a three dimensional non-singular Helmholtz boundary element method in the frequency domain.It allows for the straightforward use of higher order surface elements on the boundaries of the problem.It will been shown that the effect of the monopole sources ends up on the right hand side of the resulting matrix system.Some carefully selected examples are studied,such as point sources near and within a concentric spherical core-shell scatterer(with theoretical verification),near a curved focusing surface and near a multi-scale and multi-domain acoustic lens.

BOUNDARY INTEGRAL EQUATIONS FOR THE BENDING PROBLEM OF PLATES ON TWO-PARAMETER FOUNDATION

By means of Fourier integral transformation of generalized function, the fundamental solution for the bending problem of plates on two-parameter foundation is derived in this paper, and the fundamental solution is expanded into a uniformly convergent series. On the basis of the above work, two boundary integral equations which are suitable to arbitrary shapes and arbitrary boundary conditions are established by means of the Rayleigh-Green identity. The content of the paper provides the powerful theories for the application of BEM in this problem.