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

Liu, Bo; Xing, Yufeng; Wang, Wei; Yu, Weidong

doi:10.1016/j.compstruct.2015.06.064

Cited by 36 publications

(19 citation statements)

References 32 publications

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“…The Gauss-Lobatto quadrature is the Gauss integration with two endpoints fixed, which can be found in mathematics handbooks or in [19]. Here a simple introduction of it is presented to make the paper self-contained.…”

Section: The Differential Quadrature Finite Element Methodsmentioning

confidence: 99%

“…Reference [19] presented the detail of computing roots for Legendre polynomials using the recursion formula of Legendre polynomials if more than 40 roots are required.…”

Section: The Differential Quadrature Finite Element Methodsmentioning

confidence: 99%

“…Such low order schemes typically use low order basis functions and the accuracy is improved through mesh refinement. High order schemes like the hierarchical finite element method (HFEM) [18,19], the radial basis functions (RBFs) [20,21], the mesh free methods [22,23], the differential quadrature method (DQM) [24], and more recently, the iso-geometric analysis (IGA) [25] and the differential quadrature method finite element method (DQFEM) [26][27][28], successively emerged as highly accurate numerical methods. All works via high order methods yield excellent results due to the use of the high-order or global basis functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

Liu

Ferreira

Xing

et al. 2016

Composite Structures

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Section: The Differential Quadrature Finite Element Methodsmentioning

confidence: 99%

“…Reference [19] presented the detail of computing roots for Legendre polynomials using the recursion formula of Legendre polynomials if more than 40 roots are required.…”

Section: The Differential Quadrature Finite Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

Liu

Ferreira

Xing

et al. 2016

Composite Structures

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“…Each basis attains its maximum value on the corresponding node, and the function value varies over a small range of approximately [−0.2, 1]. In addition, this range is independent of the number of nodes; hence, very high‐order approximation is possible using these bases . For more information about Lagrange node collocation on surfaces and bodies, please refer to other works .…”

Section: Numerical Implementationmentioning

confidence: 99%

“…In addition, this range is independent of the number of nodes; hence, very high-order approximation is possible using these bases. 40 For more information about Lagrange node collocation on surfaces and bodies, please refer to other works. [41][42][43] In the present work, since the transformed bases (in natural coordinates) for w and w n on element boundaries are the one-dimensional C 2 and C 1 Hermite bases (see Figure 17), respectively, the optimized nodes that have small numerical oscillations are no longer GL nodes.…”

Section: Node Collocationmentioning

confidence: 99%

Hierarchical p‐version C¹ finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates

Xing

Liu

2019

Numerical Meth Engineering

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Summary This work focuses on the construction of p‐version finite elements that have curve boundaries for C1 problems. Both triangular and quadrilateral elements are constructed based on the C1‐version blending function interpolation methods that are developed in this work and in the literature. Orthogonal hierarchical bases are constructed and subsequently transformed into interpolative nodal bases to facilitate the imposition of boundary conditions and the implementation of C1 conformity on curvilinear domains. Nodal collocation strategies are also studied for improving the numerical performance, and novel nonuniformly distributed nodes, namely, Gauss‐Jacobi (GJ) points, are proposed. For parallelograms and straight‐sided triangular elements, C1 continuity is exactly satisfied between neighboring elements. The difficulty of C1 conformity for elements that have curved boundaries is circumvented by interpolating the normal derivatives at Gauss‐Lobatto nodes. Moreover, with the help of the blending function interpolation method, the bases on edges and the internal modes can differ in terms of approximation order. Therefore, local p‐refinements can be easily performed by these elements. Numerical results demonstrated that these elements are computationally inexpensive and converge fast for problems with regular and irregular domains.

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Strain gradient differential quadrature finite element for moderately thick micro‐plates

Zhang

Kong

et al. 2020

Numerical Meth Engineering

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Summary In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C1‐type four‐node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro‐plates. This element is free of shape functions and shear locking. The C1‐continuity requirements of deflection and rotation functions are accomplished by a fourth‐order differential quadrature (DQ)‐based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss‐Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non‐rectangular domains is proceeded by the natural‐to‐Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro‐plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro‐plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro‐plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner‐Fox‐Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.

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

Cited by 36 publications

References 32 publications

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

Hierarchical p‐version C¹ finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates

Strain gradient differential quadrature finite element for moderately thick micro‐plates

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

Cited by 36 publications

References 32 publications

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

Hierarchical p‐version C1 finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates

Strain gradient differential quadrature finite element for moderately thick micro‐plates

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Product

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Hierarchical p‐version C¹ finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates