Efficient Quadrature-Based Superconvergent Isogeometric Analysis With Macro-Integration Cells And Applications

Isogeometric analysis often employs the non-uniform rational B-splines(NURBS)used in computer aided design,as the basis functions for the geometric description as well as the subsequent finite element analysis.This methodology seamlessly integrates the computer aided geometry design and the finite element analysis.The convex and smooth NURBS basis functions offer particular advantages for the isogeometric frequency analysis.Moreover,the isogeometric high order mass matrix formulation further enhance the accuracy and convergence rates in isogeometric frequency analysis.Nonetheless,the superconvergence of higher order mass matrix approaches usually depend on the wave propagation angle,namely,the multidimensional high order mass matrices can not produce the frequency superconvergence simultaneously.Meanwhile,the numerical implementation of isogeometric higher order mass matrices is not very convenient.In this thesis,a quadrature-based superconvergent isogeometric method is presented for efficient and accurate frequency analysis with macro-integration cells and quadratic basis functions.The proposed method is capable of achieving synchronous superconvergence for all frequencies,even in multidimensional cases.In this thesis,the quadrature rule for 1D single element high order mass matrix is firstly established.It is shown that a 3-point quadrature rule enables the frequency superconvergence in 1D case.Furthermore,the smooth advantage of isogeometric basis functions is utilized to develop superconvergent quadrature rules based upon macro-integration cells.Through exactly integrating the corresponding higher order mass matrix as well as the stiffness matrix,the two-element and three-element macro-integration cells are proposed to construct explicit superconvergent quadrature rules with the 6th order frequency accuracy,which is two-order higher than that of the standard isogeometric formulation with consistent mass matrix.As a result,3-point,5-point and 7-point quadrature rules with identical precision are designed for the single element,two-element and three-element macro-integration cells,respectively.Subsequently,the 2D and 3D superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1D integration algorithms.It is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence without the wave propagation direction dependence problem,which needs special treatments for the multidimensional higher order mass matrix formulation.It is also noted that from the efficiency point of view,various macro-integration cells can be flexibly combined to discretize a given problem domain for they have identical accuracy.Numerical examples fully demonstrate the superiority of the proposed quadrature-based superconvergent isogeometric method.Finally,the proposed approach is successfully extended to the acoustic analysis with very favorable solution accuracy.