Research On Fast Multipole Boundary Element Method With High Order Elements For Analyzing Thin-walled And V-notched Structures

High precision and high efficiency have always been the goals of most engineering numerical methods.At present,constant elements are mostly used in the fast multipole boundary element method(FMBEM).However,for the physical fields with the high gradient variation and the large-scale structures with the complex geometric shapes,the FMBEM with constant elements must rely on using dense meshes and much CPU time to get the acceptable accuracy.To promote the computational efficiency,a new FMBEM with the linear and quadratic elements for analyzing the potential and elastic problems is proposed in this thesis.The algorithms to tackle the nearly singular integrals occurred in analyzing the thin-walled structures and the stress singularity at the V-notch tip are studied,so as to upgrade the capability of the proposed FMBEM with higher order elements.The main contributions of the present thesis are as follows:(1)Based on linear and three-node quadratic elements,a new FMBEM is proposed for two dimensional(2D)potential problems.An accurate and fast method is proposed to calculate the singular integrals on the elements closed to the source point.By using the forms of boundary integrals on the complex plane,the nearly singular integrals on linear elements are calculated by a complete analytical scheme and those on three-node quadratic elements are evaluated by a robust semi-analytical algorithm.Together with the fast multipole expansions for the integrals on the elements far from the source point,all boundary integrals are efficiently evaluated on the complex plane.Numerical examples show that the present FMBEM possesses higher accuracy and efficiency than the conventional FMBEM with constant elements for the same degrees of freedom.(2)A new FMBEM with linear and three-node quadratic elements is proposed for analyzing 2D elastic problems.As in the case of 2D potential problems,all boundary integrals in the FMBEM for 2D elasticity are evaluated on the complex plane,even if the boundary integrals for elastic problems are more complicate than those for potential problems.The formulations of the integral kernels are simplified by using the complex notation so that the operation process of calculating the nearly strongly and hypersingular integrals is more concise and efficient than that in the real domain.For the slender beam in bending and the stress concentration problems of the plate with multiple holes,the proposed FMBEM with higher order elements can use fewer elements to obtain the accurate results,which has a significant advantage over the conventional FMBEM with constant elements.(3)An efficient method is developed to determine the eigen-solutions of the elastic–plastic stress fields around the plane V-notch/crack tips in power-law hardening materials.The stress and displacement fields around the V-notch tip are first expressed by a series of asymptotic expansions.Then the governing ordinary differential equations(ODEs)with stress and displacement eigen-functions are established based on the fundamental equations of the elastic-plastic theory.Finally,the interpolating matrix method is employed to solve the resulting nonlinear and linear ODEs.Consequently,the complete hierarchies of eigen-solutions are obtained successively.The present method has the advantages of greater versatility and high accuracy,which is capable of dealing with the V-notches with arbitrary opening angles(including cracks)in plane strain and plane stress problems.(4)A novel highly-accurate FMBEM is proposed for analyzing the complete stress fields of the large-scale plane V-notch/crack structures.Based on the eigen-solutions of singularity problems for the linear elastic V-notch,a new multi-order singular element(MOSE)is developed to model the stress singularities in the notch/crack tip regions.The proposed FMBEM coupled with MOSEs is used to analyze the stress fields of the V-notch structures.In comparison with the existing singular elements and singularity separation techniques,the present method has evident advantages: 1)The present method possesses high accuracy because the first singular term and other higher order terms in the asymptotic solutions of stresses and displacements in the V-notch/crack tip region are considered;2)The present method is not limited by the size of opening angle of the Vnotch and is suitable for both single and bonded material problems;3)The present method has the flexibility so that the dense meshes in the tip region and reconstructing equations are not necessary,which does not impose a heavy burden in computation.(5)Based on three-node linear and six-node quadratic triangular surface elements,a new FMBEM is proposed for analyzing three dimensional(3D)potential problems.A semi-analytic scheme is introduced to calculate the nearly singular integrals in the 3D FMBEM,which makes the proposed FMBEM suitable for 3D thin-walled structures.Furthermore,a 3D FMBEM code is developed to analyze the steady-state heat transfer in thermal barrier coatings(TBCs)on the turbine blade surface.Due to accurate evaluations of the nearly singular integrals,the proposed FMBEM with higher order elements can simulate the sharp variations on temperature fields inside TBCs,which provides an accurate and efficient numerical technique for analyzing the large-scale thermal insulation performance of TBCs.In contrast to the conventional FMBEM with constant elements,the proposed FMBEM with higher order elements can use fewer degrees of freedom to achieve the same or even higher accuracy,which saves the computed time and memory size.It is worth emphasizing that the computed time and memory size of the proposed FMBEM(including 2D and 3D cases)increase almost linearly with increase of degrees of freedom as expected.The proposed FMBEM with higher order elements is applicable to analyze the large-scale thin-walled structures and V-notch/crack structures,which improves the ability of BEM to solve practical engineering problems.