Research On Bi-directional Near Singular Integration Algorithm For Boundary Element Analysis Of Elastic Problems Of Thin Structures

Elastic mechanics is an engineering problem that researchers pay close attention to today.The dominant numerical analysis method for solving this problem is finite element method(FEM),which is a global numerical method.therefore,the whole integral domain needs to be discretized in calculation,while boundary element method(BEM)only needs to discretize the boundary of the integral domain,and the Kelvin basic solution of elasticity with conventional Gaussian integral formula are used in the domain,which belongs to semi-analytical and semi-numerical method.According to the analysis of elastic mechanics problems of thin-walled structures,the geometric conditions are too harsh because the thickness of the structure is too small.The commonly used finite element method needs to rely on a large number of Gaussian integral points to maintain the ideal calculation accuracy,but it still cannot ensure the stability of calculation.The unique dimension reduction and order reduction performance of the boundary element method effectively makes up for the shortcomings of the finite element method.However,due to the inherent singularity of the basic solution,this method will produce the calculation problem of near singular integral in practical application,so it can accurately and effectively eliminate the integral near singularity of the boundary integral equation as three dimensions.This paper is devoted to dealing with the near singular integral problem more accurately and effectively,and mainly completes the following research work:(1)Most of the existing nonlinear transformation methods only pay attention to the near singularity of the integral in the radial direction,ignoring the influence of the angle direction and the shape of the integral unit.When the projection point approaches the boundary of the integral unit infinitely,it cannot obtain satisfactory calculation accuracy and is very sensitive to the shape of the integral unit.Therefore,this paper proposes three combined nonlinear transformation methods,which are iterative Sinh-Sigmoidal,iterative Sinh-Angular and bi-directional iterative Sinh,based on adaptive block technology and different coordinate transformations,and deduces their theoretical formulas to prepare for the subsequent algorithm programming.(2)C++ language is used to program the above three combined nonlinear transformation methods on VS platform.Taking the triangular integral element of surface as an example,three combined algorithms are compared with the single radial iterative Sinh algorithm,and two common numerical examples of near singular integral verify the effectiveness and accuracy of the combined algorithm.(3)The iterative Sinh-Sigmoidal algorithm with good calculation accuracy and stability is programmed and embedded into BEM program,and the elastic statics analysis of four different types of thin structures is carried out.Finally,the calculation results are compared with those obtained by the element subdivision method,which further verifies the feasibility and high accuracy of the combined algorithm.