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.

A State Space Boundary Element Method with Analytical Formulas for Nearly Singular Integrals

A new method named the state space boundary element method （SSBEM） is estab- lished, in which the problem domain is divided into two parts. One is the boundary element domain which includes the interested inner point, and the other is the state space domain. The boundary integral equation and the state space equation are combined together based on the interfacial continuity condition to form the system equation of the SSBEM. The SSBEM synthe- sizes both advantages of the boundary element method and the state space method. However, it can give inaccurate results when being used to evaluate the mechanical quantity of a point very close to the boundary element, because the Gaussian quadrature fails to calculate the nearly singular integrals. The analytical formulas were developed by part of the authors before intro- duced to deal with the nearly singular integrals. Thus, the SSBEM can yield accurate physical quantities for the points very close to the boundary element. The SSBEM results agree well with those of the finite element method （FEM）, while the discretized elements are far fewer than those of the FEM. Meanwhile, the SSBEM can analyze very thin coating, while the FEM fails due to the limitation of tolerance for Boolean operations.

EVALUATION OF SINGULAR AND NEARLY SINGULAR INTEGRALS IN THE BEM WITH EXACT GEOMETRICAL REPRESENTATION＊

The geometries of many problems of practical interest are created from circular or ellip- tic arcs. Arc boundary elements can represent these boundaries exactly, and consequently, errors caused by representing such geometries using polynomial shape functions can be removed. To fully utilize the geometry of circular boundary, the non-singular boundary integral equations （BIEs） and a general nonlinear transformation technique available for arc elements are introduced to remove or damp out the singular or nearly singular proper- ties of the integral kernels. Several benchmark 2D elastostatic problems demonstrate that the present algorithm can effectively handle singular and nearly singular integrals occur- ring in the boundary element method （BEM） for boundary layer effect and thin-walled structural problems. Owing to the employment of exact geometrical representation, only a small number of elements need to be divided along the boundary and high accuracy can be achieved without increasing other more computational efforts.

Nearly Singular Integrals in 3D Stokes Flow

A straightforward method is presented for computing three-dimensional Stokes flow,due to forces on a surface,with high accuracy at points near the surface.The flowquantities arewritten as boundary integrals using the free-spaceGreen’s function.To evaluate the integrals near the boundary,the singular kernels are regularized and a simple quadrature is applied in coordinate charts.High order accuracy is obtained by adding special corrections for the regularization and discretization errors,derived here using local asymptotic analysis.Numerical tests demonstrate the uniform convergence rates of the method.

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.

Reduction of the Regularization Error of the Method of Regularized Stokeslets for a Rigid Object Immersed in a Three-Dimensional Stokes Flow

We focus on the problem of evaluating the velocity field outside a solid object moving in an incompressible Stokes flow using the boundary integral formulation.For points near the boundary,the integral is nearly singular,and accurate computation of the velocity is not routine.One way to overcome this problem is to regularize the integral kernel.The method of regularized Stokeslet(MRS)is a systematic way to regularize the kernel in this situation.For a specific blob function which is widely used,the error of the MRS is only of first order with respect to the blob parameter.We prove that this is the case for radial blob functions with decay propertyφ(r)=O(r−3−α)when r→∞for some constantα>1.We then find a class of blob functions for which the leading local error term can be removed to get second and third order errors with respect to blob parameter.Since the addition of these terms might give a flow field that is not divergence free,we introduce a modification of these terms to make the divergence of the corrected flow field close to zero while keeping the desired accuracy.Furthermore,these dominant terms are explicitly expressed in terms of blob function and so the computation time is negligible.