Thickness-shear vibration analysis of rectangular quartz plates by a numerical extended Kantorovich method

Liu, B.; Xing, Yufeng; Eisenberger, Moshe; Ferreira, A.J.M.

doi:10.1016/j.compstruct.2013.08.021

Cited by 16 publications

(10 citation statements)

References 19 publications

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“…For simplicity of denotation, the number of sampling points on the edge in the n-and g-directions is taken as the same and is denoted by N e , and the number of hierarchical shape functions in the two directions is also taken as the same and is denoted by N h . The natural frequency is expressed by frequency parameter X used by Liu et al [17,20]. Its expression is…”

Section: Resultsmentioning

confidence: 99%

“…2) is analyzed first for validating the accuracy of the DQHFEM for thickness-shear vibrations, since there are plenty such results in literatures. For easiness of comparison, the dimensions and material constants of the rectangular quartz crystal plate are taken the same as the one used by Liu et al [17,20]. The dimensions are a = 2.…”

Section: Resultsmentioning

confidence: 99%

“…In Table 1, the DQHFEM results are compared with the DQFEM results [20], 3D FEM results [23] and results obtained by a numerical extended Kantorovich method (NEKM) [17]. Both DQFEM and NEKM used 5 Â 5 elements.…”

Section: Resultsmentioning

confidence: 99%

“…Both DQFEM and NEKM used 5 Â 5 elements. The NEKM [17] used the separation variable technique and is very efficient but may lose accuracy. Both DQHFEM and DQFEM are two dimensional methods of high accuracy.…”

Section: Resultsmentioning

confidence: 99%

“…After all, since the scalar differential equation can be easily separated in Cartesian coordinates, free vibration frequencies and modes of unelectroded rectangular AT-cut plates can be obtained through many approximate techniques [6,12]. Wang et al [13] used the Kantorovich method [14] and the analytical solution procedure of [15,16] (NEKM) [17] was proposed for thickness-shear vibration of rectangular crystal plates that iteratively reduced the two-dimensional elasticity problem into two one-dimensional ones and then solved them using a differential quadrature finite element method (DQFEM) [18,19]. The DQFEM was also directly applied to thickness-shear vibration of rectangular crystal plates by Liu and Xing [20].…”

Section: Introductionmentioning

confidence: 99%

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Section: Resultsmentioning

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

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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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The transverse vibration of Mindlin rectangular plates with internal elastic supports and arbitrary boundary supports

et al. 2023

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Thickness-shear vibration analysis of rectangular quartz plates by a numerical extended Kantorovich method

Cited by 16 publications

References 19 publications

Thickness-shear vibration analysis of circular quartz crystal plates by a differential quadrature hierarchical finite element method

Thickness-shear vibration analysis of circular quartz crystal plates by a differential quadrature hierarchical finite element method

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

The transverse vibration of Mindlin rectangular plates with internal elastic supports and arbitrary boundary supports

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