Superconvergent Isogeometric Frequency Analysis For Elastic Waves

Isogeometric analysis unifies the computer aided design and the finite element method through employing the same non-uniform rational B-splines(NURBS)in both geometric description and finite element analysis.This methodology eliminates the geometric discretization error and significantly improves the computational accuracy.The convex and smooth NURBS basis functions lead to superior performance for the frequency computation associated with dynamic problems.In the frequency analysis of scalar wave equations,it has been shown that the superconvergent quadrature rules can be designed for both mass and stiffness matrices.Along this path,a quadrature-based superconvergent isogeometric method is developed in this thesis to accurately and efficiently compute the frequencies for elastic waves.Firstly,the isogeometric discretization of elastic wave problem is described and a concise method to derive the accuracy measure for elastic wave frequencies is established.Based upon this method,the frequency accuracy with the standard 2D isogeometric consistent mass matrix formulation is investigated.Subsequently,2D superconvergent quadrature rules for the isogeometric analysis of elastic wave frequencies are devised,which elevate the frequency accuracy by two orders compared with the isogeometric consistent mass matrix formulation.Through a natural tensor product operation,the 2D quadrature rules are generalized to 3D cases in a straightforward manner,which lead to a 3D superconvergent computation of elastic wave frequencies.Finally,the proposed quadrature rules are further simplified for the purpose of efficient practical implementation.A series of numerical examples congruously demonstrate that the proposed quadrature-based superconvergent isogeometric method is capable of achieving very favorable accuracy for the frequency analysis of elastic waves.