Free Flexural Vibration Analysis of Thin Plates Using NURBS-Augmented Finite-Element Method

Abstract

In isogeometric analysis (IGA), the non-uniform rational B-spline (NURBS) basis functions are used for depicting the geometry and the displacement field. As the NURBS basis functions are non-interpolating in nature, the enforcement of essential boundary condition becomes a difficult task. In order to circumvent the above problem, recently the authors Mishra and Barik (Comput 232:105869, 2020; Eng Comput 35:351–362, 2019) proposed a new method called NURBS-augmented finite-element method (NAFEM). The authors have incorporated the non-uniform rational B-spline (NURBS) basis functions for the representation of the geometry and the usual finite-element basis functions are adopted for the field variables as they satisfy the Kronecker-Delta property. This simplifies the implementation of the boundary condition to a great extent. In the present work, NAFEM is extended for free vibration analysis of plates having different geometries and boundary conditions, and the results are found to be in excellent agreement with the existing ones. To showcase the robustness of NAFEM, some arbitrary-shaped plates have also been considered, and the new results are presented.

Introduction

The finite-element analysis (FEA) is applied to many disciplines where engineering structures are concerned. The structure’s geometry plays a significant role in the analysis and hence its accurate representation is inevitable for more realistic analysis. The challenge faced by the researchers to deal with the analysis of plates of various geometrical configurations steered to develop many methods and techniques. Some researchers proposed techniques for dealing with multiple planforms when some of the investigators proposed to analyze a particular shape of the plate.

A semi-analytic approach is suggested for free vibration analysis of annular sector plates [3]. The subparametric concept is used in the spline finite-strip method to examine plates of general shape for static and free vibration analyses [4]. Vibration analysis of plates of general quadrilateral and sectorial planforms is carried out by applying the differential quadrature method using the geometric mapping technique [5]. Kirchhoff plates of arbitrary shape have been studied for free vibration using the mapping technique in the finite-element method [6, 7].

The free vibration analysis of plates of different geometries is presented by Lee [8] using a four-noded plate element. He has considered the natural strains based on Reissner–Mindlin assumptions taking account of the shear deformation and rotatory inertia effect. Shear-deformable plates of different geometrical configurations for free vibration and buckling analyses have been reported employing the mesh-free method based on the reproducing kernel particle approximate [9]. The vibration of several structural models such as rods, thin beams, membranes, and thin plates are studied by Cottrell et al. [10] using isogeometric analysis (IGA). The authors have considered the rotationless beams and plates as three-dimensional solid models and used the knot refinement concept to get more accurate and robust results than the corresponding finite elements. Reali [11] applied the concept of IGA to study the response of one-dimensional and two-dimensional problems under vibration.

The amalgamation of the subparametric triangular plate bending element with first-order shear deformation theory for the analysis of a square plate with a circular cut-out at the center is described in [12]. The NURBS-enhanced finite-element method (NEFEM) [13] is an improvement to the classical finite-element method. This method can represent the geometry precisely through computer-aided design (CAD) description of the boundary with NURBS. In this technique, the elements which do not intersect the edge require a standard finite-element interpolation function and numerical integration for the analysis. In contrast, the elements which cross the NURBS boundary require a specially designed piecewise polynomial interpolation function and numerical integration. The authors have studied the application of NEFEM to 2D Poisson problems and electromagnetic scattering simulation showing its advantages compared to the classical isoparametric finite-element formulation. A family of elements of smooth, curved geometry using rational Bézier functions for boundary description has been introduced by Lu [14] by which the common shapes such as circles and ellipses are described precisely. These elements are suited for analyzing discrete bodies undergoing more or less uniform or regular deformation.

Irregular shapes thin plates for free vibration have been analyzed by quadrature element method (QEM) [15, 16]. Free vibration analysis of annular sector plates and graphene sheets is performed using eight-node curvilinear domains in discrete singular convolution (DSC) method [17, 18].

The direct imposition of inhomogeneous essential boundary conditions to the NURBS control points is found to be problematic, leading to significant errors with deteriorated rates of convergence [19]. Therefore, the investigators have represented an improved formulation for NURBS-based isogeometric analysis by employing a transformation method to the boundary points, which relates the control variables to the collocated nodal values at the essential boundary. Using the open knot vectors, the resulting NURBS basis functions associated with control points vanish at the periphery. Bazilevs et al. [20] explored T-splines, which is a generalization of NURBS enabling local refinement, as a basis for IGA. The researchers have applied bivariate and trivariate T-splines of various degrees to elementary fluid and structural mechanics problems. Hughes et al. [21] have initiated the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb known as the half-point rule has emerged, indicating that optimal rules involve several points roughly equal to half the number of degrees-of-freedom or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order of the basis.

The Dirichlet boundary conditions in IGA can be suitably imposed by taking the quasi-interpolation methods into account [22]. de Falco et al. [23] have developed GeoPDEs, which is a suite of free software tools for applications of IGA focusing on providing a common framework for the implementation of the many IGA methods for the discretization of currently studied partial differential equations, mainly based on B-splines and NURBS. Different geometric Kirchhoff plates have been undertaken for free vibration analysis by a moving Kriging interpolation-based mesh-free method [24]. Schmidt et al. [25] have presented a methodology enabling IGA on trimmed NURBS surfaces. A local reconstruction technique using a geometric basis has been developed and applied to evaluate the finite-element constituents of the trimmed knot spans in terms of the underlying control variables. The Lagrange multiplier method to impose the essential boundary conditions for improving the accuracy of the solution in the IGA of thin plates has been used in [26].

The isogeometric static, dynamic, and buckling analyses of rectangular and circular-shaped functionally graded material (FGM) plates for different boundary conditions have been carried out by Tran et al. [27]. Here, higher order shear deformation theory (HSDT) model is developed using $C^{1}$ continuous elements to improve the accuracy of the solution and taking the stress distribution without using shear correction factors. Blending NURBS with Lagrangian representations in IGA has been first developed by Lu et al. [28]. In the blended representation, selected boundary edges or surfaces of a multivariate NURBS patch are parametrized in (rational) Lagrangian form. The Lagrangian parameters are obtained by transforming the NURBS representation. The transformation, by construction, exactly preserves the original geometry, which is helpful in interfacing the NURBS domain to finite element domains or for imposing essential boundary conditions.

The bending, free flexural vibration, buckling, and flutter behavior of square and skew functionally graded material (FGM) plates under different boundary conditions using NURBS-based finite element is being studied by Valizadeh et al. [29]. Further, the isogeometric finite-element analysis is incorporated with refined plate theory to study the behavior of functionally graded material square and circular plates for static, free vibration and buckling analyses in [30] where displacement field is approximated with four degrees-of-freedom per each control point allowing an efficient solution process. The isogeometric locking free plate formulation for the bending, buckling, and free vibration analyses of homogeneous and functionally graded square, circular and square plates with complicated cut-out considering different boundary conditions and gradient index is adopted in [31]. Jüttler et al. [32] have initiated G+SMO (Geometry+Simulation Modules), an open-source, C++ library for IGA. It is an object-oriented template library that implements a generic concept for IGA, based on abstract classes for discretization basis, geometry map, assembler, solver, and using the object polymorphism and inheritance techniques to provide a common framework of IGA for a variety of different available basis types. Nguyen and Nguyen-Xuan [33] have proposed an efficient computational tool based on isogeometric finite-element formulation of three-dimensional elasticity for static and dynamic response of functionally graded square, circular, annular, and square plates with complicated shaped cut-outs. The numerical tests have shown that a quartic NURBS element can eliminate the shear-locking phenomena. An effective formulation combining the extended-isogeometric approach and higher order shear deformation theory for free vibration analysis of cracked functionally graded material plates is presented by Tran et al. [34]. This formulation accounts for the effects of gradient index, crack-length, crack-location, length-to-thickness ratio on the natural frequencies and mode shapes of simply supported and clamped FGM plates.

Vázquez [35] has presented a new design for the implementation of IGA in Octave and MATLAB package for the solution of partial differential equations. Compared to the previous version [23], the new design is more efficient in terms of memory consumption and computational time. Massarwi and Elber [36] have developed tools for approximation and local re-parameterization of trimmed elements for three-dimensional problems based on volumetric modeling via volumetric representations (V-reps). A differential quadrature hierarchical finite-element method (DQHFEM) is proposed for vibration, and bending analyses of Mindlin plates with curvilinear domains [37]. The non-uniform rational Lagrange (NURL) alternative basis, which are interpolation functions to represent NURBS geometries for IGA, has been reported in [38]. The authors have incorporated the same basis functions to carry out the in-plane and flexural vibration of thin plates to overcome the difficulties of IGA using NURBS on coping with Dirichlet boundary conditions. Free vibration analysis of conical and cylindrical shells and annular plates made of composite, functionally graded materials (FGM), and carbon nanotube reinforced (CNTR) composite using FSDT via discrete singular convolution method is carried out in [39]. A comprehensive review on trimming in IGA has been showcased in [40] in the context of design, data exchange, and computational simulation. Employing the Fourier expansion method, sector-like thin plates having simply supported radial edges have been analyzed for transverse vibration adopting a semi-analytical approach [41]. Antolin et al. [42] have demonstrated a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries. Improved Fourier series method (IFSM) is used by Liu et al. [43] for free in-plane vibration of arbitrarily shaped straight-sided quadrilateral and triangular plates. They have solved the problems by mapping the arbitrarily shaped plates into a unit square plate and following the usual modeling of vibration problems for rectangular plates. Alihemmati and Beni [44] have proposed a mesh-free Galerkin method for analysis of plates of triangular and polygonal geometries. Recently, Sahoo and Barik [45,46,47] have analyzed curved and straight-edged stiffened plates for free vibration and dynamic response to moving loads. They have employed an isoparametric finite element with shear deformation to accomplish this task.

The aero-thermo-elastic panel flutter response of a functionally graded plate having cracks in the supersonic flow field has been analyzed by Khalafi and Fazilati [48]. The authors have incorporated IGA along with Nitsche technique which based is on FSDT. The parametric instability response of laminated composite plates under uniform in-place loading has been carried out for the first time in [49]. Here, the authors have incorporated the IGA-based FEM formulation for addressing the instability of the panels. The response of a perforated flat panel having an externally bonded two-steered patch of variable stiffness has been presented in [50]. Khalafi and Fazilati [51] studied the free vibration of a repaired perforated plate. They have incorporated the FSDT of plates for the multi-patch modeling approach. The free vibration and linear flutter analyses of laminated square and skew shaped plates have been carried out in [52]. The formulation is based on the FSDT in-conjunction with an aerodynamic loading model. Free vibration analysis of laminated composite plates having curved perforations has been extensively reported in [53]. The researchers have considered an IGA formulation with Nitsche method using FSDT. Liu et al. [54] proposed the in-plane free vibration of arbitrary plates having various end conditions using Improved Fourier Series Method (IFSM). Here, the authors have mapped the plates to a unit square plate and the usual procedure adopted in case of rectangular plates is used for the solution. A NURBS-based multi-patch IGA formulation with FSDT has been presented in [55]. Do and Lee [56] studied the free vibration response of FGM plates with cut-outs by incorporating IGA. Here, the authors have formulated a quasi-3D higher order shear deformation theory(HSDT).

The researchers continuously endeavor to have analytical solutions to the arbitrary shape thin-plate analysis. But the analytical expressions become too complex and unmanageable when it is attempted for plates having curved edges and other than rectangular or straight-sided planforms. In the numerical method of analysis, FEM was predominant so far in formulating new elements to model the arbitrary shape of the plates. Isoparametric element was considered a successful one in dealing with the non-rectangular plate geometries, though it is deficient in describing the exact curved edges. Moreover, as this element is based on Mindlin’s theory, the shear strain term leads to the shear-locking problem when applied to thin plates and needs special attention of reduced integration.

There was a significant change of approach when the Computer-Aided Design (CAD) was developed to represent the geometry accurately, and this was embedded into FEA [57] through the concept of IGA, where the researchers employed the NURBS basis functions for representation of the geometry as well as the field variables required for the analysis. As the NURBS basis functions are non-interpolating in nature, the enforcement of essential boundary condition was found to be difficult and seeks special treatment [58,59,60,61,62,63,64].

In the present work, the above mentioned boundary condition imposition problem is alleviated by replacing the NURBS with classical finite-element basis function to represent the field variables. The NURBS basis functions are only used for describing the geometry, and the four-noded ACM (Adini, Clough and Melosh [65]) plate bending element having 12 degrees of freedom is considered for the displacement field. The interpolating nature of the ACM plate bending element helps in enforcing the essential boundary condition similar to the classical finite element method, making the whole process simpler. The NAFEM has been successfully applied to analyses of arbitrarily shaped plates [1, 2, 66, 67]. The present work is an extension of NAFEM for analyzing plates of different planforms under various end conditions for free vibration.

Problem Statement

The equation of equilibrium for an elastic system under free vibration in matrix form is given by

$$\begin{aligned} \left[ K\right] \left\{ \delta \right\} +\left[ M\right] \left\{ \ddot{\delta }\right\} =\left\{ 0\right\} , \end{aligned}$$

(1)

where $\left[ K\right]$ is the global elastic-stiffness matrix, $\left[ M\right]$ is the global consistent mass matrix, $\left\{ \delta \right\}$ is the displacement vector, $\left\{ \ddot{\delta }\right\}$ is the acceleration vector. Equation (1) can be solved using the standard matrix analysis once the element matrices are assembled into the global matrices.

Finite-Element Formulation

NURBS Basis Function

Given a knot vector $S=\begin{Bmatrix} s_{1},s_{2},s_{3},...s_{n+p+1} \end{Bmatrix}$, the associated set of B-spline basis functions $\begin{Bmatrix} N_{i,p} \end{Bmatrix}_{i=1}^{n}$ are defined recursively by the Cox–de-Boor formula [68], starting with the zeroth order basis function $(p=0)$ as

$$\begin{aligned} N_{i,0}=\left\{ \begin{array}{ll} 1 &{} \text {if }s_{i}\le s\le s_{i+1}\\ 0 &{} \text {otherwise}, \end{array}\right. \end{aligned}$$

(2)

and for a polynomial order $p\ge 1$

$$\begin{aligned}&N_{i,p}(s)=\frac{s-s_{i}}{s_{i+p}-s_{i}}N_{i,p-1}(s)\nonumber \\&\quad +\frac{s_{i+p+1}-s}{s_{i+p+1}-s_{i+1}}N_{i+1,p-1}(s), \end{aligned}$$

(3)

where n is the number of basis functions and p is the order of the basis functions. The fractions of the form 0/0 are defined as zero.

Mapping of the Plate

The use of NURBS basis functions for representation of geometry introduces the concept of parametric space which is absent in the conventional finite-element formulation [68]. The consequence of this additional space is that an additional mapping is performed to operate in the parent element coordinates. First, the parent space is mapped to the parametric space and then to the physical space (Fig. 1).

Fig. 1

Mapping between domains/spaces

The mapping from parametric to physical space is given by

$$\begin{aligned} \left[ \begin{matrix} X \end{matrix}\right] =\sum _{i=1}^{n}\sum _{j=1}^{m}P_{i,j}R_{i,j}^{p,q}(s,t), \end{aligned}$$

(4)

where p and q are the order, n and m are the number of control points, $P_{i,j}$ are the control points and $R_{i,j}^{p,q}(s,t)$ is the bivariate NURBS basis function defined as

$$\begin{aligned}&R_{i,j}^{p,q}(s,t)=\frac{N_{i}(s)M_{j}(t)w_{i,j}}{\sum \nolimits _{\bar{i}=1}^{n}\sum \nolimits _{\bar{j}=1}^{m}N_{\bar{i}}(s)M_{\bar{j}}(t)w_{\bar{i},\bar{j}}}, \end{aligned}$$

(5)

where ${N_{i}(s)}$ and $M_{j}(t)$ are the univariate B-spline basis functions of order p and q corresponding to the knot vectors in the respective directions and $\begin{Bmatrix} w_{i,j} \end{Bmatrix}_{i=1,j=1}^{n,m}$, where $w_{i,j}>0$ are the set of NURBS weights. The mapping from parent to parametric space is given by [68]

$$\begin{aligned}&s(\xi )=\frac{\left( s_{i+1}-s_{i}\right) \xi +\left( s_{i+1}-s_{i}\right) }{2} \end{aligned}$$

(6)

$$\begin{aligned}&t(\eta )=\frac{\left( t_{i+1}-t_{i}\right) \eta +\left( t_{i+1}-t_{i}\right) }{2}. \end{aligned}$$

(7)

For complex geometries the physical space may be divided into simple patches and then those patches are mapped to the parent space through the parametric space. As a typical example the physical space defined by the coordinates ACDB is divided into two patches namely ACFE and EFDB as shown in Fig. 2. The parent space is then mapped to the patches in the physical space through the parametric space and the procedure is followed as before to compute the stiffness matrix of each patch. The stiffness matrices of all the patches are assembled to form the global stiffness matrix of the whole plate. Thus the analysis of the plates having complicated geometry can be made simpler by subdividing them into more amenable patches which can be dealt with ease [1].

Fig. 2

Mapping of complex shaped geometry through different domains/spaces

Mesh Generation

The mesh for the plate is generated using the knot refinement technique of [68]. In this, the plate geometry is generated first using the NURBS (Fig. 3a), and then applying the h-refinement technique [23, 68], the desired meshes can be obtained. Figure 3 shows the different stages of mesh generation for a typical semi-circular semi-elliptical plate which has also been used as an example problem in this paper. For detail of this technique, Section 2.1.4 of [69] may be referred.

Fig. 3

Mesh generation of a typical semi-circular semi-elliptical plate

Displacement Interpolation Function

For the proposed element, the four-noded rectangular non-conforming ACM plate bending element with 12 degrees-of-freedom is taken as the basic element. As the element is in $\xi$-$\eta$ plane the shape functions and the nodal parameters for the displacements and slopes are expressed in terms of the coordinates $\xi -\eta$ unlike the $x-y$ coordinates of the parent ACM element [6]. The shape functions for the displacement field for the $j\mathrm{th}$ node are given as

$$\begin{aligned}&\left[ \begin{matrix} N_{w}^{j}&N_{\theta _{\xi }}^{j}&N_{\theta _{\eta }}^{j} \end{matrix}\right] =\frac{1}{8}\nonumber \\&\times \quad \left[ \begin{matrix} (s_{0}+1)(t_{0}+1)\left( 2+s_{0}+t_{0}-\xi ^{2}-\eta ^{2}\right) &{} \\ s_{j}(s_{0}+1)^{2}(s_{0}-1)(t_{0}+1) &{} \\ t_{j}(s_{0}+1)(t_{0}+1)^{2}(t_{0}-1) \end{matrix}\right] ^{T}, \end{aligned}$$

(8)

where $s_{0}=\xi s_{j}$ and $t_{0}=\eta t_{j}$.

The detailed formulation is presented in Section 2.4 of [1, 2].

Strain–Displacement Matrix

The strain–displacement matrix is given by $[B]=[T][\bar{B}]$, where $[\bar{B}] =\left[ \begin{array}{ccccc} \dfrac{\partial N_{w}}{\partial \xi }&\dfrac{\partial N_{w}}{\partial \eta }&\dfrac{\partial ^{2} N_{w}}{\partial \xi ^{2}}&\dfrac{\partial ^{2} N_{w}}{\partial \eta ^{2}}&\dfrac{\partial ^{2} N_{w}}{\partial \xi \,\partial \eta } \end{array}\right] ^\mathrm{T}$ and $[T] =$ $\left[ \begin{matrix} [T_{F_{1}}]&[T_{F_{2}}] \end{matrix}\right]$; $[T_{F_{1}}]=-[J6]^{-1}[J5][J4]^{-1}[J1]^{-1}$ , $[T_{F_{2}}]=[J6]^{-1}[J3]^{-1}.$

The detail formulation of the above can be referred from Section 2.5 of [1, 2].

Stiffness Matrix

Total potential energy of the plate element is given by

$$\begin{aligned} \Pi _{p}=\dfrac{1}{2}\iint \begin{Bmatrix} \epsilon (x,y) \end{Bmatrix}^\mathrm{T}\begin{Bmatrix} \sigma (x,y) \end{Bmatrix} \mathrm{d}x\; \mathrm{d}y-\iint w^\mathrm{T} \mathrm{d}x\; \mathrm{d}y. \end{aligned}$$

(9)

Applying the principle of minimum potential energy and making appropriate substitutions for strain vector ($\begin{Bmatrix} \epsilon (x,y) \end{Bmatrix}$), the stress vector ($\begin{Bmatrix} \sigma (x,y) \end{Bmatrix}$) and the displacement field (w—Eq. (8) of [1]), we have

$$\begin{aligned}{}[K]_{e}\begin{Bmatrix} \delta \end{Bmatrix}=\begin{Bmatrix} P \end{Bmatrix}_{e}, \end{aligned}$$

(10)

where $\begin{Bmatrix} \delta \end{Bmatrix}$ is the vector of nodal displacements, $\begin{Bmatrix} P \end{Bmatrix}_{e}$ is the vector of nodal forces and $[K]_{e}$ is the plate element stiffness matrix given by

$$\begin{aligned}{}[K]_{e}=\iint [B]^\mathrm{T}[D][B]\, \mathrm{d}x\, \mathrm{d}y. \end{aligned}$$

(11)

Since the NURBS basis is a function of s and t, Eq. (11) becomes

$$\begin{aligned}{}[K]_{e}=\iint [B]^\mathrm{T} [D][B]\begin{vmatrix} J4 \end{vmatrix}\, \mathrm{ds}\, \mathrm{d}t. \end{aligned}$$

(12)

Again since [B] is a function of $\xi$ and $\eta$, Eq. (12) becomes

$$\begin{aligned}{}[K]_{e}=\iint [B]^\mathrm{T}[D][B]\begin{vmatrix} J1 \end{vmatrix}\begin{vmatrix} J4 \end{vmatrix}\, \mathrm{d}\xi \, \mathrm{d}\eta . \end{aligned}$$

(13)

The integration is carried out numerically by adopting $2\times 2$ Gaussian quadrature formula.

Consistent Mass Matrix

Reproducing the procedures adopted in [7], the consistent mass matrix of the plate element is formulated on the basis of the lateral displacement w. The acceleration of a point in the middle plane of the plate in terms of the interpolation function given in Sect. 2.4 of [1, 2] can be expressed as

$$\begin{aligned} \begin{Bmatrix} \ddot{f} \end{Bmatrix}=\begin{Bmatrix} \ddot{w} \end{Bmatrix}=\left[ N_{w}\right] \begin{Bmatrix} \ddot{\delta } \end{Bmatrix}. \end{aligned}$$

(14)

Hence, the inertia force for a small volume $\mathrm{d}V$ at that point is given by

$$\begin{aligned} \begin{Bmatrix} f_{I} \end{Bmatrix}=\rho \, \mathrm{d}V\,\left[ N_{w}\right] \begin{Bmatrix} \ddot{w} \end{Bmatrix}=\rho \, \mathrm{d}V\,\left[ N_{w}\right] \begin{Bmatrix} \ddot{\delta } \end{Bmatrix}. \end{aligned}$$

(15)

where $\rho$ is the mass density of the plate material. If $\left\{ F_{I} \right\}$ is the nodal inertia force parameter, then the contribution of inertia in the equation of motion can be obtained from the principle of virtual work and can be expressed as

$$\begin{aligned}&\left\{ \mathrm{d}\delta ^\mathrm{T}\right\} \left\{ F_{I}\right\} =\int _{v}\left\{ \mathrm{d}f^\mathrm{T} \right\} \left\{ f_{I} \right\} . \end{aligned}$$

(16)

The above equation with the help of Eq. (15) can be rewritten as

$$\begin{aligned}&\left\{ \mathrm{d}\delta ^\mathrm{T} \right\} \left\{ F_{I}\right\} \nonumber \\&\quad = \int _{v}\{ d\delta ^{T} \} \left[ N_{w}\right] \nonumber \\&\quad \rho \,dV\{ \ddot{\delta }\} \end{aligned}$$

(17)

from which

$$\begin{aligned}&\{ F_{I} \} =\rho \int _{v}\left[ N_{w}\right] ^\mathrm{T}\nonumber \\&\quad \left[ N_{w}\right] \mathrm{d}V\{ \ddot{\delta }\}\nonumber \\&\quad =[M]_{e}\{ \ddot{\delta }\}, \end{aligned}$$

(18)

where $[M ]^{e}$ is the mass matrix of the bare plate element and for constant thickness t, it is given by

$$\begin{aligned}&[M ]_{e}=\rho \int _{v}[N_{w}]^\mathrm{T}[N_{w}]\, \mathrm{d}V =\rho \,t\int \int [N_{w}]^\mathrm{T}[N_{w}]\,\mathrm{d}x\,\mathrm{d}y\nonumber \\&\quad =\rho \,t\int \int [N_{w}]^\mathrm{T}[N_{w}]\,\begin{vmatrix} J4 \end{vmatrix}\,\mathrm{d}s\,\mathrm{d}t\nonumber \\&\quad =\rho \,t\int \int [N_{w}]^\mathrm{T}[N_{w}]\,\begin{vmatrix} J1 \end{vmatrix}\begin{vmatrix} J4 \end{vmatrix}\,\mathrm{d}\xi \,\mathrm{d}\eta . \end{aligned}$$

(19)

Boundary Conditions

Following the procedure similar to the case of static and buckling analysis [1, 2], the stiffness of the boundary is given by

$$\begin{aligned}{}[K_{b}]=\int [N_{b}]^T[N_{k}]\begin{vmatrix} J_b \end{vmatrix}\,\mathrm{d}\lambda _1, \end{aligned}$$

(20)

where

$$\begin{aligned}{}[N_{b}]=\begin{bmatrix} 1 &{} 0 &{} 0 \\[10pt] 0 &{} \cos \beta &{} \sin \beta \\[10pt] 0 &{} -\sin \beta &{} \cos \beta \end{bmatrix}\begin{Bmatrix} [N_w] \\[8pt]\dfrac{\partial [N_w]}{\partial x}\\[10pt] \dfrac{\partial [N_w]}{\partial y}\end{Bmatrix} \end{aligned}$$

(21)

$$\begin{aligned} =\begin{bmatrix} k_w &{} 0 &{} 0 \\[10pt]0 &{} k_\alpha \cos \beta &{} k_\alpha \sin \beta \\[10pt]0 &{} -k_\beta \sin \beta &{} k_\beta \cos \beta \end{bmatrix}\begin{Bmatrix} [N_w] \\[10pt] \dfrac{\partial [N_w]}{\partial x}\\[8pt] \dfrac{\partial [N_w]}{\partial y}\end{Bmatrix} \end{aligned}$$

(22)

and Jacobian,

$$\begin{aligned} \begin{vmatrix} J_{b} \end{vmatrix}= \dfrac{\mathrm{d} s_{1}}{\mathrm{d} \lambda _{1}}. \end{aligned}$$

(23)

This boundary stiffness matrix contributes to that element to which the boundary belongs to.

Solution Procedure

The solution procedure adopted in the free vibration analyses of plates are presented in this section. The equilibrium equation for the free vibration is given by Eq. (1). Considering the motion as harmonic motion, the solution of Eq. (1) is

$$\begin{aligned} \left\{ \delta \right\} =H\left\{ \psi \right\} e^{i\omega t}, \end{aligned}$$

(24)

where $\left\{ \psi \right\}$ is a normalized vector of the order of $\left\{ \delta \right\}$, H is the weighting parameter of $\left\{ \psi \right\}$ and $\omega$ is the natural frequency of vibration in radians per second. On substitution, the equilibrium equation becomes

$$\begin{aligned} \left[ K\right] \left\{ \psi \right\} =\omega ^{2}\left[ M\right] \left\{ \psi \right\} . \end{aligned}$$

(25)

This is a generalized eigenvalue problem and is solved by the simultaneous iteration algorithm of [70] and its solution is the eigenvalue $\omega ^{2}$ and the eigenvector $\left\{ \psi \right\}$. The skyline [71] technique is adopted for the storage of global elastic-stiffness matrix $\left[ K\right]$ and the mass matrix $\left[ M\right]$. In this process of storage, the matrix is stored in a single array eliminating the zero entries if any within the band thus reducing the storage requirement of the computer.

Numerical Examples

The free vibration analyses of a number of plates having different shapes and boundary conditions are carried out using the simultaneous iteration algorithm of [70] and the results obtained are compared with the existing ones. The results are presented in tabular form with a mesh division of $32\times 32$ for the whole plate. Figures of typical plates showing mesh divisions of $8\times 8$ along with the nodes (asterisk) are presented for each case. The abbreviations used in the table for the boundary conditions (S—simply supported, C—clamped, and F—free) are depicted in the anti-clockwise direction starting from the left edge of the plate.

Rectangular Plates

The convergence study for the different mesh sizes for the simply supported and clamped rectangular plate of aspect ratio 1 (square plate) is presented in Table 1 where excellent convergence of results is obtained with increasing mesh divisions of the plate. The free vibration results of rectangular plates having aspect ratios 0.4, 1.0, 1.5 and 2.5 and under different support conditions are presented in Table 2. The results are compared with that of analytical method [72] and they are found to be in excellent agreement. First ten natural frequency results of a simply supported rectangular plate having different thicknesses have been presented in Table 3. In another case, the effect of thickness to width ratio (h/b) on the natural frequencies of antisymmetric modes of a rectangular plate have has presented in Table 4. The present results are in excellent agreement with Classical Plate Theory (CPT [73]) and that of Lim et al. [74]. The variation of the natural frequency with respect to the change in thickness to breadth (h/b ratio) is found to be negligible. A typical rectangular plate having $8\times 8$ mesh is shown in Fig. 4. It is observed that the frequency parameter increases with the increase in aspect ratio. The frequency parameter is found to be more for the all edges clamped plate in comparison to other support conditions. The first four mode shapes for rectangular plates under simply supported and clamped conditions are shown in Figs. 5, 6, 7, 8, 9, 10, 11 and 12.

Table 1 Convergence study for dimensionless natural frequencies $\left( \beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}} \right)$ of a rectangular plate under different boundary condition (aspect ratio ${a}/{b}=1$)

Table 2 Dimensionless natural frequencies $\left( \beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}} \right)$ of a rectangular plate under different boundary condition

Fig. 4

A typical rectangular plate with having $8\times 8$ mesh

Table 3 First ten natural frequencies of a simply supported rectangular plate having different thickness (Young’s modulus $E = 70\times 10^9$ $\mathrm{N} /\mathrm{m}^2$, density $\rho =2700$ $\mathrm{kg}/\mathrm{m}^3$, Poisson’s ratio $\nu =0.3$, length of plate $=$ 0.6 $\mathrm{m}$, breadth of plate $=$ 0.4 $\mathrm{m}$)

Table 4 Effect of thickness ratio (h/b) on the natural frequencies of antisymmetric modes of rectangular plates

Fig. 5

Mode 1 of rectangular plate having aspect ratio $a/b=1.0$ for CCCC boundary condition

Fig. 6

Mode 2 of rectangular plate having aspect ratio $a/b=1.0$ for CCCC boundary condition

Fig. 7

Mode 3 of rectangular plate having aspect ratio $a/b=1.0$ for CCCC boundary condition

Fig. 8

Mode 4 of rectangular plate having aspect ratio $a/b=1.0$ for CCCC boundary condition

Fig. 9

Mode 1 of rectangular plate having aspect ratio $a/b=1.0$ for SSSS boundary condition

Fig. 10

Mode 2 of rectangular plate having aspect ratio $a/b=1.0$ for SSSS boundary condition

Fig. 11

Mode 3 of rectangular plate having aspect ratio $a/b=1.0$ for SSSS boundary condition

Fig. 12

Mode 4 of rectangular plate having aspect ratio $a/b=1.0$ for SSSS boundary condition

Free Vibration of Circular Plates

The free vibration analysis of circular plates (Fig. 13) with three different boundary conditions (completely free, simply supported and clamped) are carried out and the results compared in Tables 5, 6 and 7 with the published ones are found to be in well agreement. It is observed that the dimensionless natural frequencies of a fully clamped circular plate is more in comparison to a simply supported and completely free circular plates.

Fig. 13

A typical circular plate with having $8\times 8$ mesh

Table 5 Dimensionless natural frequencies $\left( \beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}} \right)$ of a completely free circular plate

Table 6 Dimensionless natural frequencies $\left( \beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}}\right)$ of a fully clamped circular plate

Table 7 Dimensionless natural frequencies $\left( \beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}} \right)$ of a simply supported circular plate

Free Vibration of Skew Rhombic Plates

Skew rhombic plates (Fig. 14) having different skew angles ($\alpha$ anti-clockwise from y-axis, aspect ratio a/b = 1.0) and support conditions are analyzed and the results are presented in Tables 8, 9 an 10 with a mesh division of $32\times 32$ and the results are compared with those of [6, 58] and they agree well. From the results, it is observed that the frequency parameter for a skew rhombic plate increases with the skew angle and is highest under fully clamped condition. The first four mode shapes of skew rhombic plate having skew angle $\theta =30^{\circ }$ for simply supported and SCSC boundary conditions are presented in Figs. 15, 16, 17, 18, 19, 20, 21 and 22.

Table 8 Frequency parameters $\lambda = \omega a^{2}\left( \rho \,h/D\right) ^{\frac{1}{2}}$ for skew rhombic plate

Table 9 Frequency parameters $\lambda = \omega a^{2}\left( \rho \,h/D\right) ^{\frac{1}{2}}$ for skew rhombic plate

Table 10 Frequency parameters $\lambda = \omega a^{2}\left( \rho \,h/D\right) ^{\frac{1}{2}}$ for skew rhombic plate

Fig. 14

A typical skew rhombic plate having $8\times 8$ mesh

Fig. 15

Mode 1 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SSSS boundary condition

Fig. 16

Mode 2 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SSSS boundary condition

Fig. 17

Mode 3 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SSSS boundary condition

Fig. 18

Mode 4 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SSSS boundary condition

Fig. 19

Mode 1 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SCSC boundary condition

Fig. 20

Mode 2 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SCSC boundary condition

Fig. 21

Mode 3 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SCSC boundary condition

Fig. 22

Mode 4 of skew plate having aspect ratio $a/b=1.0$ and skew angle $\theta = 30^{\circ }$ under SCSC boundary condition

Annular Sector Plate

The free vibration analysis of an annular sector plate (Fig. 23) having sector angle $\alpha =90^{\circ }$, with different aspect ratios ($r_\mathrm{i}/r_\mathrm{o}$, $r_\mathrm{i}$—inner radius, $r_\mathrm{o}$—outer radius) and under different boundary conditions is carried out. The results obtained are presented in Tables 11, 12 are found to be in excellent agreement with Barik [7] than Mukhopadhyay [3]. It is observed that the frequency parameter for an annular sector plate is highest under fully clamped condition. The first four mode shapes of annular sector plates having aspect ratio $r_\mathrm{i}/r_\mathrm{o}=0.25$ under CSSS and simply supported boundary conditions are presented in Figs. 24, 25, 26, 27, 28, 29, 30 and 31. The frequency parameter increases with the increase in the aspect ratio $r_\mathrm{i}/r_\mathrm{o}$ and is found to be maximum for fully clamped end condition in comparison to others.

Fig. 23

A typical annular plate with $8\times 8$ mesh

Table 11 Frequency parameters $\lambda = \omega a^{2}\left( \rho \,h/D\right) ^{\frac{1}{2}}$ for clamped annular sector plate

Table 12 Frequency parameters $\lambda = \omega a^{2}\left( \rho \,h/D\right) ^{\frac{1}{2}}$ for annular sector plate

Fig. 24

Mode 1 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under CSSS boundary condition

Fig. 25

Mode 2 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under CSSS boundary condition

Fig. 26

Mode 3 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under CSSS boundary condition

Fig. 27

Mode 4 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under CSSS boundary condition

Fig. 28

Mode 1 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under SSSS boundary condition

Fig. 29

Mode 2 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under SSSS boundary condition

Fig. 30

Mode 3 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under SSSS boundary condition

Fig. 31

Mode 4 of annular sector plate having $r_\mathrm{i}/r_\mathrm{o}=0.25$ under SSSS boundary condition

L-Shaped Plate

The free vibration analysis of L-shaped plate (Fig. 32) under different boundary conditions is performed and the results obtained are presented in Tables 13, 14 which agree well with the existing ones. The minor variation of results can be attributed to the manner in which the boundary condition are imposed in each case. This issue of implementation of the support conditions is well documented in Shojaee et al. [58]. It is observed that the dimensionless natural frequencies of a fully clamped L-shaped plate is more in comparison to a simply supported and completely free one.

Fig. 32

A typical L-shaped plate with $8\times 8$ mesh

Table 13 Dimensionless natural frequencies $\beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}}$ of a completely free L-shaped plate

Table 14 Dimensionless natural frequencies $\beta _{1}=(\omega ^{2}a^{4}\rho h/D)^{\frac{1}{4}}$ of a L-shaped plate

Semi-circular Semi-elliptical Plate

The free vibration analysis of a plate consisting of a semi-circle and semi-ellipse (Fig. 33) is carried out for different boundary conditions and aspect ratios b/a (where a is the radius of the semi-circle and the semi-minor axis of the semi-ellipse and b is the semi-major axis of the semi-ellipse) and the results are presented in Table 15. The boundary conditions are depicted with C—clamped, S—simply supported and F—free end in the anti-clockwise direction starting from the left edge of the plate in the following sequence AB–BD–DE–EA. It is observed that the dimensionless natural frequencies of a fully clamped semi-circular semi-elliptical plate is highest in comparison to other ends. The first four mode shapes of semi-circular semi-elliptical plate having aspect ratio $a/b=1.125$ under clamped and CSFF boundary conditions are presented in Figs. 34, 35, 36, 37, 38, 39, 40 and  41.

Fig. 33

A typical semi-circular semi-elliptical plate with $8\times 8$ mesh

Fig. 34

Mode 1 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CCCC boundary condition

Fig. 35

Mode 2 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CCCC boundary condition

Fig. 36

Mode 3 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CCCC boundary condition

Fig. 37

Mode 4 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CCCC boundary condition

Fig. 38

Mode 1 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CSFF boundary condition

Fig. 39

Mode 2 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CSFF boundary condition

Fig. 40

Mode 3 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CSFF boundary condition

Fig. 41

Mode 4 of semi-circular semi-elliptical plates having aspect ratio $a/b=1.125$ under CSFF boundary condition

Table 15 Dimensionless natural frequencies $\beta _{1}=\omega ^{2}b^{2}(\rho h/D)^{\frac{1}{2}}$ of semi-circular semi-elliptical plate

Rectangular Plate with Curved Edges

A rectangular plate having curved edges (Fig. 42) is analyzed for different boundary conditions and aspect ratios $r_{1}/r_{2}$ (where $r_{1}$ and $r_{2}$ are the semi-major and semi-minor axes of the plate respectively) and the results are presented in Table 16. The boundary conditions are depicted with C—clamped, S—simply supported and F—free end in the anti-clockwise direction starting from the left edge of the plate in the following sequence AB–BD–DE–EA. It is observed that the natural frequency for a rectangular plate with curved edges increases with the increase in the aspect ratios $r_{1}/r_{2}$ and is highest for fully clamped edge condition. First four mode shapes of the plate having aspect ratio $r_1/r_2=1.5$ under clamped and CCSS boundary conditions are shown in Figs. 43, 44, 45, 46, 47, 48, 49 and  50.

Fig. 42

A typical rectangular plate with curved edges having $8\times 8$ mesh

Fig. 43

Mode 1 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 44

Mode 2 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 45

Mode 3 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 46

Mode 4 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 47

Mode 1 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCSS boundary condition

Fig. 48

Mode 2 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCSS boundary condition

Fig. 49

Mode 3 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCSS boundary condition

Fig. 50

Mode 4 of rectangular plate with curved edges having aspect ratio $r_1/r_2=1.5$ under CCSS boundary condition

Table 16 Dimensionless natural frequencies $\beta _{1}=\omega ^{2}r_{1}^{2}(\rho h/D)^{\frac{1}{2}}$ of rectangular plate with curved edges

Dome-Shaped Plate

A typical plate resembling the shape of a dome is considered by taking one of the edges straight and its opposite edge as the top of a dome (Fig. 51). The free vibration analysis of this dome-shaped plate is carried out for different boundary conditions and aspect ratio $r_{1}/r_{2}$ where $r_{1}$ is half the length of the straight edge and $r_{2}$ is the semi-minor axis of the dome. The results obtained are presented in Table 17. The boundary conditions are depicted with C—clamped, S—simply supported and F—free end in the anti-clockwise direction starting from the left edge of the plate in the following sequence EA–AB–BD–DE. It is observed that the natural frequency for a dome-shaped plate increases with the increase in the aspect ratios $r_{1}/r_{2}$ and is highest for fully clamped edge condition. First four mode shapes of the plate having aspect ratio $r_1/r_2=1.5$ under different clamped and CCCS boundary conditions are shown in Figs. 52, 53, 54, 55, 56, 57, 58 and  59. The dimensionless natural frequency increases with the increase in aspect ratio and was found to be maximum for all edges clamped condition.

Fig. 51

A typical dome-shaped plate having $8\times 8$ mesh

Fig. 52

Mode 1 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 53

Mode 2 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 54

Mode 3 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 55

Mode 4 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCC boundary condition

Fig. 56

Mode 1 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCS boundary condition

Fig. 57

Mode 2 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCS boundary condition

Fig. 58

Mode 3 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCS boundary condition

Fig. 59

Mode 4 of a plate in the shape of a dome having aspect ratio $r_1/r_2=1.5$ under CCCS boundary condition

Table 17 Dimensionless natural frequencies $\beta _{1}=\omega ^{2}r_{1}^{2}(\rho h/D)^{\frac{1}{2}}$ of a dome-shaped plate

Rectangular Plate with One-Side Curved Edge

A rectangular plate with one side being curved is analyzed by considering the rectangular portion as patch-1 ($32\times 32$ mesh) and the remaining portion as patch-2 ($32\times 32$ mesh) under different boundary conditions and aspect ratios a/b (a—length of the rectangular portion, b—breadth of the rectangular portion). The nodes of the rectangular portion (patch-1) are represented with asterisk and the remaining portion (patch-2) with circular markers respectively. The results obtained are presented in Table 18. A typical rectangular plate with one-side curved edge consisting of a rectangular portion as patch-1 ($4\times 4$mesh) and the remaining as patch-2 ($4\times 4$ mesh) is shown in Fig. 60. It is observed that the natural frequency for a rectangular plate with one-side curved edge increases with the increase in the aspect ratios a/b and is highest for fully clamped edge condition.

Table 18 Dimensionless natural frequencies $\beta _{1}=\omega a^{2}(\rho h/D)^{\frac{1}{2}}$ of a rectangular plate with one-side curved edge

Fig. 60

A typical rectangular plate with one-side curved edge consisting of a rectangular portion as patch-1 ($4\times 4$ mesh) and the remaining as patch-2 ($4\times 4$ mesh)

Conclusions

In the present formulation, NURBS basis functions are used to represent the exact shape of the arbitrary thin plates. In contrast to the isogeometric analysis, the use of classical finite-element basis functions as field variables helps in imposing the boundary conditions easily which is the main drawback of isogeometric analysis. Further, the knot refinement technique of the NURBS basis function takes care of the mesh generation. The free vibration analysis of arbitrary-shaped plates are carried out and the results obtained are found to be well in agreement with the published ones. To showcase the robustness of NAFEM, some complicated shaped plates (semi-circular semi-elliptical, rectangular plate with curved edges, dome-shaped plate, and L-shaped plate) have been considered for the free vibration analysis and the new results are presented. Further, a rectangular plate with one side being curved is analyzed by considering the rectangular portion as patch-1 and the remaining portion as patch-2 thereby showing the capability of NAFEM to analyze complex geometries by splitting them into more amenable patches which can be dealt with ease.