Research On Several Kinds Of Singular Problems By Boundary Element Method

With the rapid development of computer technology,CAE analysis has been widely used in the field of engineering.In the CAE technology,there are two main numerical methods: finite element method(FEM)and boundary element method(BEM).As an engineering method,the FEM is widely used in the mechanical industry,but it has some inherent defects.The BEM is a semi-analytic method that has the advantages of high stress accuracy and dimensionality reduction,and therefore has natural advantages when dealing with singularity problems.The boundary integral equation is the basis of the BEM.The singular problems studied in this paper include two aspects:(1)singularities arising in the boundary integral equations,including quasi-singular domain integrals,strong singular integrals,and hyper-singular integrals;(2)problems with singular stresses solved by BEM,including crack problems and V-notch problems.The fundamental solution in the boundary integral equation is singular,which brings some difficulties to the application of the BEM.But it is the singularity of the fundamental solution that guarantees the stability of the numerical analysis of the BEM.Therefore,it is very crucial to correctly handle the singularity of the fundamental solution in the boundary integral equation.Firstly,a solution for quasi-singular domain integrals is proposed in this paper.Then an intrinsic coordinate approximation expansion method is presented for the strong singular and hyper-singular integrals.Finally,the BEM is used to solve the crack problem and V-notch problem.The main work and research results of this paper are as follows:(1)The solutions for two-dimensional(2D)and three-dimensional(3D)quasi-singular domain integrals are proposed.When the pseudo-initial condition method is used to solve the transient heat conduction problems,the domain integral with the time-dependent fundamental solution needs to be calculated.This paper first studies the characteristics of the time-dependent fundamental solution.That is,when the time step is small,the domain integral with the time-dependent fundamental solution has similar properties of the singular integral.Then 2D quasi-singular domain integrals are processed by nonlinear transformations such as polar transformation,(α,β)transformation and Sinh transformation,respectively,and the(α,β)transformation is found to be the best.Finally,the solutions for 2D and 3D quasi-singular domain integrals are determined.For 2D problems,(α,β)transformation coupled with cell subdivision method is used.While for 3D problem,the(α,β,γ)transformation coupled with 3D cell subdivision method is adopted,and by introducing the concept of the closest point,the problem that the source point located outside the element is solved.(2)The dual-interpolation boundary element method(DiBEM)is used to evaluate the stress intensity factor of the 2D crack and to simulate crack propagation.In this paper,the strong and hyper-singular integrals are first calculated by using intrinsic coordinate approximation expansion method.Then the shape functions of the singular element based on the DiBEM are deduced.Finally the stress intensity factors are evaluated by using the crack open displacements.The crack propagation direction is determined by the maximum circumferential stress criterion,and the crack growth rate is calculated by Paris law.The single edge inclined crack and the double edge crack propagation path are simulated.Numerical examples show that the accuracy of the stress intensity factor calculated by the DiBEM is higher than that obtained by the traditional BEM when the number of the source points in these two methods is equal.(3)For the vertex singularity,a singular point element is proposed,and the stress intensity factor of the 3D crack is evaluated by the method of the combination of the dual boundary element method(DBEM)and the singular point element.This paper first introduces the basic idea of the DBEM and deduces the boundary integral equation of displacement discontinuity method.Meanwhile,the order of vertex singularity is studied,and it is concluded that in most cases,the order of vertex singularity approaches-0.5.Based on this conclusion,the shape function of the singular point element is deduced.Finally,the stress intensity factor of 3D cracks is calculated by using the method of combining DBEM and singular point element.Numerical examples show that the displacement field on the free surface can be simulated more accurately by applying the singular point element,so that it is not necessary to arrange a dense mesh near the singular point.(4)By studying the order of stress singularity of V-shaped notch,a novel singular element for V-shaped notch is proposed.The V-notch problem is different from the crack problem in that its stress has multiple singularities and varies with the notch angle.By analyzing the variation of the first and second eigenvalues with the notch angle,we find that the stress singularity is mainly determined by the first eigenvalues,especially when the notch angle is large.Based on this characteristic,a new singular element for V-shaped notch is proposed.The displacement field around the notch tip can be more accurately simulated with the proposed singular element.Therefore the stress intensity factor can be evaluated more accurately.