Analysis of elastostatic problems using a semi-analytical method with diagonal coefficient matrices

A new semi-analytical method is proposed for solving boundary value problems of two-dimensional elastic solids, in this paper. To this end, the boundary of the problem domain is discretized by specific non-isoparametric elements that are proposed for the first time in this research. These new elements employ higher-order Chebyshev mapping functions and new special shape functions. For these shape functions, Kronecker Delta property is satisfied for displacement function and its derivatives. Furthermore, the first derivatives of shape functions are assigned to zero at any given control point. Eventually, implementing a weak form of weighted residual method and using Clenshaw–Curtis quadrature, coefficient matrices of equations system become diagonal, which results in a set of decoupled governing equations to be used for solving the whole system. In other words, the governing equation for each degree of freedom (DOF) is independent from other DOFs of the problem. Validity and accuracy of the present method are fully demonstrated through four benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.

► The boundary of the domain is discretized by specific non-isoparametric elements.
► The new elements use Chebyshev mapping functions and new special shape functions.
► A weak form of weighted residual method and Clenshaw–Curtis quadrature is used.
► Coefficient matrices of equations system are diagonal.
► The governing equation for each DOF is independent from other DOFs of the problem.