A superconvergent isogeometric method is developed for the buckling analysis of thin beams and plates, in which the quadratic basis functions are particularly considered. This method is formulated through refining the quadrature rules used for the numerical integration of geometric and material stiffness matrices. The criterion for the quadrature refinement is the optimization of the buckling load accuracy under the assumption of harmonic buckling modes for thin beams and plates. The method development starts with the thin beam buckling analysis, where the material stiffness matrix with quadratic basis functions does not involve numerical integration and thus the refined quadrature rule for geometric stiffness matrix can be obtained in a relatively easy way. Subsequently, this refined quadrature rule for thin beam geometric stiffness matrix is conveniently generalized to the thin plate geometric stiffness matrix via the tensor product operation. Meanwhile, the refined quadrature rule for the thin plate material stiffness matrix is derived by minimizing the buckling load error. It turns out that the refined quadrature rule for the thin plate material stiffness matrix generally depends on the wave numbers of buckling modes. A theoretical error analysis for the buckling loads evinces that the isogeometric method with refined quadrature rules offers a fourth-order accurate superconvergent algorithm for buckling load computation, which is two orders higher than the standard isogeometric analysis approach. Numerical results well demonstrate the superconvergence of the proposed method for the buckling loads corresponding to harmonic buckling modes, and for those related with non-harmonic modes, the buckling loads given by the proposed method are also much more accurate than their counterparts produced by the conventional isogeometric analysis.

The fourth-order governing equation of Euler-Bernoulli beams necessitates the employment of C1 Hermite shape functions in finite element analysis. However, unlike the C0 finite element shape functions that can be easily formulated as Lagrangian polynomials in a unified manner, the Hermite shape functions of Euler-Bernoulli beam elements are usually constructed case by case. In this work, a simple and unified meshfree path is proposed to develop arbitrary order Hermite shape functions of Euler-Bernoulli beam elements. This approach is realized by proving that the Hermite reproducing kernel meshfree shape functions degenerate to the interpolatory Hermite finite element shape functions under a proper choice of support sizes. The proposed approach provides an explicit formalism for Hermite finite element shape functions, which enables an easy construction of arbitrary order Hermite beam elements. Accordingly, in addition to the conventional cubic Hermite beam elements, explicit shape functions, mass and stiffness matrices for quintic and septic Hermite beam elements are presented. Meanwhile, the frequency errors and general higher order mass matrix formulation for these Hermite beam elements are elaborated as well. These theoretical results are consistently demonstrated by numerical examples with respect to both static and free vibration analysis.

An optimization of quadrature rules is presented for the isogeometric frequency analysis of wave equations using cubic splines. In order to optimize the quadrature rules aiming at improving the frequency accuracy, a frequency error measure corresponding to arbitrary four-point quadrature rule is developed for the isogeometric formulation with cubic splines. Based upon this general frequency error measure, a superconvergent four-point quadrature rule is found for the cubic isogeometric formulation that achieves two additional orders of frequency accuracy in comparison with the sixth-order accuracy produced by the standard approach using four-point Gauss quadrature rule. One interesting observation is that the first and last integration points of the superconvergent four-point quadrature rule go beyond the domain of conventional integration element. However, these exterior integration points pose no difficulty on the numerical implementation. Subsequently, by recasting the general four-point quadrature rule into a three-point formation, the proposed frequency error measure also reveals that the three-point Gauss quadrature rule is unique among possible three-point rules to maintain the same sixth-order convergence rate as the four-point Gauss quadrature rule for the cubic isogeometric formulation. These theoretical results are clearly demonstrated by numerical examples.

A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines, which reveals an interesting coarse mesh superconvergence. Firstly, the frequency error estimates for isogeometric discretization of Mindlin–Reissner plates with quadratic splines are rationally derived, where the degeneration to Timoshenko beams is discussed as well. Subsequently, in accordance with these frequency error measures, the shear locking issue corresponding to the full integration isogeometric formulation is elaborated with respect to the frequency accuracy deterioration. On the other hand, the locking-free characteristic for the isogeometric formulation with uniform reduced integration is illustrated by its superior frequency accuracy. Meanwhile, it is found that a frequency superconvergence of sixth order accuracy arises for coarse meshes when the reduced integration is employed for the isogeometric free vibration analysis of shear deformable beams and plates, in comparison with the ultimate fourth order accuracy as the meshes are progressively refined. Furthermore, the mesh size threshold for the coarse mesh superconvergence is provided as well. The proposed theoretical results are consistently proved by numerical experiments.

A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.