Study Of Nearly Singular Integrals In Boundary Element Methods With Two-order Elements And Its Applications To3-D Acoustic Problems

The application of boundary element method in engineering is confined due tothe evaluation difficulty of the nearly singular integrals for a long time. Someregularization algorithms about the nearly singular integrals on the linear elementsof BEM have been established successfully. Nevertheless, the calculation of thenearly singular integrals on the high-order geometry elements, especially inthree-dimensional BEM, is still a very difficult problem. Geometrical boundary isoften complex in engineering. In this case, high-order geometry elements will fitthe true boundary accurately. Therefore, study of singular integrals in boundaryelement methods with high-order element is necessary.The basic theory of boundary element methods is introduced firstly. Thefundamental solutions and the nearly singular of boundary integral equation ofelasticity problem, potential problem and acoustic problem are elaborated. Theregularization algorithm of the nearly singular integrals of linear elements in BEMis reviewed. Study of the nearly singular integrals of high-order geometry elementsis carried out systematically. The semi-analytical algorithms are applied to dealwith the nearly singular integrals for three-dimensional acoustic problem.In this thesis, the geometric characteristics for the high-order curve surfaceelements are analyzed under the Global coordinate、the local Cartesian coordinateand polar coordinate systems, where4-node quadrilateral curve element is taken asa sample.According to Taylor’s formula, the fundamental solutions ofthree-dimensional acoustic problem are transformed into polynomial form. Thekernel function of boundary integral equation is transformed into singular kernelfunction and non-singular kernel function by means of add and subtract items. Thenearly singular integrals are transformed into singular integrals and non-singularintegrals by subtraction method for the integrals containing singular kernel function.Non-singular integrals is calculated with the conventional Gauss method, theanalytical algorithms of the polar variable for singular integrals is calculated byseparation of variables at local polar coordinates, and then the angle variableintegral is numerically calculated with the conventional Gauss method also.The examples show that the proposed semi-analytical algorithms for the high-ordergeometry elements have the strong feasibility and high accuracy.Semi-analytical algorithms of the high-order geometry elements have generalapplicability.It can be developed to calculate the nearly singular integrals in otherphysical field.The application of boundary methods in engineering is widened.