Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper.These equations are received after applying the integral transformation and Gauss-Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium.Initially,three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate.So,the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only.The thorough analysis of both displacement and traction kernels is accomplished,and similarity in behavior of both kernels is established.The kernels are expressed in terms of complete elliptic integrals of first and second kinds.The second kind elliptic integrals are nonsingular,and standard Gaussian quadratures are applied for their numerical evaluation.Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities.The numerical treatment of these integrals takes into account the presence of this integrable singularity.The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity,Catalan’s constant,the Gaussian surface integral.The comparison between analytical and numerical data has proved high precision and availability of the proposed method.
The application of the singular boundary method(SBM),a relatively new meshless boundary collocation method,to the inverse Cauchy problem in threedimensional(3D)linear elasticity is investigated.The SBM involves a coupling between the non-singular boundary element method(BEM)and the method of fundamental solutions(MFS).The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS.Due to the boundary-only discretizations and its semi-analytical nature,the method can be viewed as an ideal candidate for the solution of inverse problems.The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique,while the optimal regularization parameter is determined by the L-curve criterion.Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.
A new numerical method,scaled boundary isogeometric analysis(SBIGA)combining the concept of the scaled boundary finite element method(SBFEM)and the isogeometric analysis(IGA),is proposed in this study for 2D elastostatic problems with both homogenous and inhomogeneous essential boundary conditions.Scaled boundary isogeometric transformation is established at a specified scaling center with boundary isogeometric representation identical to the design model imported from CAD system,which can be automatically refined without communication with the original system and keeping geometry invariability.The field variable,that is,displacement,is constructed by the same basis as boundary isogeometric description keeping analytical features in radial direction.A Lagrange multiplier scheme is suggested to impose the inhomogeneous essential boundary conditions.The new proposed method holds the semi-analytical feature inherited from SBFEM,that is,discretization only on boundaries rather than the entire domain,and isogeometric boundary geometry from IGA,which further increases the accuracy of the solution.Numerical examples,including circular cavity in full plane,Timoshenko beam with inhomogenous boundary conditions and infinite plate with circular hole subjected to remotely tension,demonstrate that SBIGA can be applied efficiently to elastostatic problems with various boundary conditions,and powerful in accuracy of solution and less degrees of freedom(DOF)can be achieved in SBIGA than other methods.
