A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains

Abstract

A differential quadrature hierarchical finite element method (DQHFEM) is proposed by expressing the hierarchical finite element method matrices in similar form as in the differential quadrature finite element method and introducing interpolation basis on the boundary of hierarchical finite element method elements. The DQHFEM is similar as the fixed interface mode synthesis method but the DQHFEM does not need modal analysis. The DQHFEM with non-uniform rational B-splines elements were shown to accomplish similar destination as the isogeometric analysis. Three key points that determine the accuracy, efficiency and convergence of DQHFEM were addressed, namely, (1) the Gauss-Lobatto-Legendre points should be used as nodes, (2) the recursion formula should be used to compute high-order orthogonal polynomials, and (3) the separation variable feature of the basis should be used to save computational cost. Numerical comparison and convergence studies of the DQHFEM were carried out by comparing the DQHFEM results for vibration and bending of Mindlin plates with available exact or highly accurate approximate results in literatures. The DQHFEM can present highly accurate results using only a few sampling points. Meanwhile, the order of the DQHFEM can be as high as needed for high-frequency vibration analysis.A DIFFERENTIAL QUADRATURE HIERARCHICAL FINITE ELEMENT METHOD 175 (i) The mesh of the structure only has to be generated once for all. The convergence is reached by increasing the p-order [6, 7] without mesh refinement [8]. Input data can be reduced to the minimum, which greatly simplifies pre-post-processing [3]. (ii) The HFEM has an embedding property, which means the stiffness and mass coefficients can be retained as the order of interpolation is increased [3,6]. This makes it possible to introduce error indicators for adaptive analysis [6,9]. (iii) Simple structures can be modeled using just one element. Therefore, the assembly of elements is avoided [3]. Complex structures can be analyzed by a few big elements. (iv) Elements of different polynomial degree can be easily joined. So it is possible to include additional degrees of freedom where needed [3]. (v) The HFEM can give accurate results with far fewer degrees of freedom than the h-version because of the use of high-order polynomial basis functions [3,[10][11][12][13]. This feature is important for nonlinear analysis using finite element method [11][12][13].The HFEM is obviously superior to the ordinary h-version FEM. The HFEM has made significant progress [10,14,15] since it was proposed about 40 years ago. Some commercial software like MSC NASTRAN already has HFEM elements [16]. However, the applications of HFEM are still far less prevalent than the h-version FEM because of both historical reasons and the problems existing in HFEM. The HFEM uses high or very high-order orthogonal polynomials as basis functions of auxiliary degrees of freedom. So the HFEM provides many opportunities as shown earlier; however, it also brings many challenges. The problems that nee...

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