Study On A Novel Regularization Algorithm Of Singular Integrals In Boundary Element Methods With High-order Elements And Its Applications

The present status of the researches for evaluation of the nearly singular integrals in boundary element methods (BEM) is investigated and reviewed. Up to now, some regularization algorithms about the nearly singular integrals on the linear elements of BEM have been established successfully. However, the calculation of the nearly singular integrals on the high-order elements, especially in three-dimensional BEM (3-D BEM), is still a very difficult problem, which has been handicapping the applications of BEM in engineering.The regularization algorithm of the nearly singular integrals on the linear elements in BEM is introduced firstly. In this thesis, the algorithm is developed to analyzing three-dimensional (3-D) acoustics problem by BEM. Then, By means of the idea of the regularization algorithm for the linear element, a kind of novel semi-analytical algorithms for the high-order elements are proposed to calculate the nearly strongly singular and hyper-singular integrals in2-D and3-D BEM, where the quadratic element is taken as a sample. And then, the present semi-analytical algorithms are performed to deal with the nearly singular integrals in the BE analysis with the high-order elements for two-dimensional (2-D) and3-D potentials and their thin domain problems,2-D elasticity and its thin-walled structures. The main contributions in the thesis can be shown in the following.1. The regularization algorithm of the nearly singular integrals on the linear element is developed to the BE analysis of3-D acoustics. In the present thesis, the fundamental solutions of3-D acoustics are expanded as the Taylor series so that the boundary integral expressions are separated as both the non-singular integral parts and the singular integral parts where the later is equivalent to the lead singular term of the fundamental solutions. Consequently, the regularization algorithm is applied to calculating the nearly strongly singular and hyper-singular integrals in the BE analysis of3-D acoustics. Some examples are shown that the present computed results are more accurate than ones of the conventional BEM. It is noted that the accurate physical quantities at the inner pointes which are very close to the boundary are valuable for the applications of acoustic analysis in engineering. As an idea, the present technique of the Taylor series expansion with respect to some functions in the fundamental solution can be generalized to solving the nearly singular integrals in other boundary integral equations where explicit forms of their fundamental solutions are not rational functions.2. A novel semi-analytical algorithm is proposed to deal with the nearly singular integrals on high-order elements in2-D BEM. In the present thesis, by the geometrical analysis for the high-order element, the relative distance from a source point to the integral element is defined as approach degree (denoted as e1) which can measure possible influence of the singularity of the integrals, where3-node quadratic curve element is taken as a sample without lost generality. Then the equivalent singular functions are separated from the integral kernels about the high-order elements in2-D potential boundary integral equation by means of an asymptotic analysis for the nearly singular integrals with respect to the local coordinate on the element. And the subtract ion strategy is applied to removing the singularities of the integral expressions. Therefore, the new semi-analytical algorithm is established, which can accurately evaluate the nearly strongly singular integrals within the range of e1>10-14and the nearly hyper-singular integrals within the range of e1>10-7. The semi-analytical algorithm has been successfully applied to the BE analysis of2-D potential and its thin domain problems, which can also obtain the inner potentials and fluxes close to the boundary.3. The idea of the above semi-analytical algorithm is expanded to the BE analysis of2-D elasticity. The same manipulation as the above is done by taking3-node quadratic curve element as a sample. In the discrete boundary integral equation with the high-order elements, the equivalent singular functions are separated from the integral kernel functions by the local coordinate system transformation. Then the nearly strongly singularity and hyper-singularity on the integral elements are removed by the use of the subtraction technique. The formulations of the semi-analytical algorithm for computing the nearly strongly singular and hyper-singular integrals are obtained in terms of tedious manipulation. The semi-analytical algorithm has been successfully employed in the multi-domain BE analysis of2-D elasticity to solving the inner displacements and stresses close to the boundary. Several benchmark numerical examples demonstrate that the present regularized algorithm about the high-order elements is more accurate and efficient than one about the linear elements in BEM as well as finite element method for analyzing the thin-walled structures and laminate structures.4. Another new semi-analytical algorithm is proposed to deal with the nearly singular integrals on high-order elements in3-D BEM. Through the geometrical analysis for the high-order curve surface element under the local Cartesian coordinate and polar coordinate systems, the relative distance from a source point to the integral element is defined as approach degree which is a main factor led to the nearly singular surface integrals, where8-node quadrilateral curve surface element is taken as a sample without lost generality. The approximate singular parts of the fundamental solutions in3-D potential boundary integral equations are separated from the integral kernel functions by means of the transformations of two local coordinate systems and the asymptotic analysis for the nearly singular integrals in the local polar coordinate. Consequently the nearly strongly singularity and hyper-singularity on the integral surface elements are eliminated by subtracting the approximate singular parts from the integral kernels instead of the direct numerical quadrature. The singular surface integrals related to the equivalent singular functions can be separately calculated about two integral variables in turn. It follows that the integral with respect to polar variable p is firstly represented with the analytical formulations and then the leading integral with respect to variable θ is numerically calculated with the conventional Gauss method, which is named as the semi-analytical algorithm by the thesis for evaluating the nearly singular integrals in3-D BEM. The present semi-analytical approach has been applied to the BE analysis of3-D potential and its thin domain problems. Both the nearly strongly and hyper-singular integrals are computed accurately.It is noted that the proposed semi-analytical algorithms can be developed, without any difficulty, to calculate the nearly strongly and hyper-singular integrals on other high-order elements in2-D and3-D BEM, which indicates that the puzzle of the evaluations of the nearly strongly and hyper-singular integrals in BEM has been solved well in the present thesis. The present algorithms have been successfully employed to the BE analysis of potential and elasticity with the high-order elements, which makes the BEM have more advantage than FEM in analyzing2-D and3-D thin body problems.