A frequency error estimation for isogeometric analysis of Kirchhoff–Love cylindrical shells

Abstract

A frequency error estimation is presented for the isogeometric free vibration analysis of Kirchhoff–Love cylindrical shells using both quadratic and cubic basis functions. By analyzing the discrete isogeometric equations with the aid of harmonic wave assumption, the frequency error measures are rationally derived for the quadratic and cubic formulations for Kirchhoff–Love cylindrical shells. In particular, the governing relationship of the continuum frequency for Kirchhoff–Love cylindrical shells is naturally embedded into the frequency error measures without the need of explicit frequency expressions, which usually are not trivial for the shell problems. In accordance with these theoretical findings, the 2nd and 4th orders of frequency accuracy are attained for the isogeometric schemes using quadratic and cubic basis functions, respectively. Numerical results not only thoroughly verify the theoretical convergence rates of frequency solutions, but also manifest an excellent magnitude match between numerical and theoretical frequency errors for the isogeometric free vibration analysis of Kirchhoff–Love cylindrical shells.

Graphical abstract

Keywords

isogeometric analysis / Kirchhoff–Love cylindrical shell / free vibration / frequency error / convergence

1 Introduction

The Kirchhoff–Love cylindrical shells, or thin shells, are frequently encountered in engineering practice, which are governed by a set of the 4th order differential equations and require at least a C^1 approximation in the Galerkin formulation [1]. The isogeometric analysis proposed by Hughes et al. [2,3] uses highly smooth B-splines, non-uniform rational B-splines (NURBSs), or their variants as the basis functions simultaneously for exact geometry representation and finite element analysis. Consequently, isogeometric analysis leads to a resurgence of employing thin plate and shell models associated with the Kirchhoff–Love assumption. For example, Kiendl et al. [4] presented an isogeometric shell analysis with accurate Kirchhoff–Love elements. Zhang et al. [5] presented a quasi-convex coupled isogeometric-meshfree method for the free vibration analysis of cracked Kirchhoff plates. A NURBS-based inverse analysis was carried out by Vu-Bac et al. [6] to reconstruct the nonlinear deformations of thin shell structures. Based on the Kirchhoff–Love shell theory and isogeometric discretization, Guo et al. [7] investigated the sensitivity of shell buckling analysis. A horizontal circular cylindrical shell partially filled with fluid was studied by Yildizdag et al. [8] using an isogeometric FE−BE approach. Thai et al. [9] proposed an isogeometric cohesive zone model for the delamination analysis of Kirchhoff–Love shells, and a bi-material topology optimization was presented by Chen et al. [10] using coupled structural-acoustic finite and boundary elements in the context of isogeometric thin shell analysis.

As for the structural vibration analysis, the superior frequency spectra of isogeometric analysis have been illustrated by Cottrell et al. [11], Reali [12], Hughes et al. [13], and other related works [14–20]. In terms of thin plate and shell vibrations, Atri and Shojaee [21] investigated the natural frequencies of thin shell problems with isogeometric analysis. Yin et al. [22] presented a rotation free isogeometric analysis based on Kirchhoff–Love theory to study the free vibration of functionally graded thin plates. Nguyen-Thanh et al. [23] examined the free vibration analysis of cracks in thin-shell structures using the consistently coupled isogeometric-meshfree approach [24,25]. Borković et al. [26] analyzed the free vibration of singly curved shells with an isogeometric finite strip method. Mohammadi et al. [27] performed the isogeometric free and forced vibrations for sinusoidally corrugated functionally graded carbon nanotube-reinforced composite thin panels. Recently, Liu et al. [28] carried out an isogeometric vibration analysis of piezoelectric Kirchhoff–Love shells with Catmull−Clark subdivision surfaces, and a Bezier extraction-based numerical implementation of isogeometric analysis was presented by Du et al. [29] for the modeling of thin-walled structures. Moreover, higher order isogeometric mass matrices have also been developed to improve the frequency accuracy for thin beam and plate problems [30].

Meanwhile, it is noted that the theoretical frequency accuracy for isogeometric free vibration analysis was presented by Cottrell et al. [11], Reali [12], and Wang et al. [14], among others, for the 2nd order continuum and 4th order thin plate problems. A detailed frequency accuracy study for the isogeometric free vibration analysis of curved Euler−Bernoulli beams was presented by Sun et al. [31]. However, due to its inherent complexity, the theoretical study for the isogeometric free vibration analysis of Kirchhoff–Love shells has not been found in the open literature. Consequently, this work aims at providing a systematic theoretical investigation on the frequency error estimation for Kirchhoff–Love cylindrical shell vibrations. Within this study, the isogeometric formulation for Kirchhoff–Love cylindrical shells is elaborated with particular reference to explicit mass and stiffness matrices regarding quadratic and cubic basis functions. To enable the development of an analytical error estimation for Kirchhoff–Love cylindrical shell vibrations, the relationship governing the continuum frequency is directly employed in the theoretical analysis, which successfully bypasses the need of solving very complex continuum frequency expressions. Subsequently, a frequency error estimation is attained for the isogeometric analysis of Kirchhoff–Love cylindrical shells, which evinces that the 2nd and 4th orders of frequency accuracy are achieved for quadratic and cubic basis functions, respectively. These theoretical accuracy measures are then well demonstrated by numerical results.

The rest of this paper is organized as follows. In Section 2, the governing equations of Kirchhoff–Love cylindrical shells and their isogeometric discretization are discussed. Section 3 presents a comprehensive theoretical investigation regarding the frequency accuracy of isogeometric free vibration analysis of Kirchhoff–Love cylindrical shells. Subsequently, the proposed analytical frequency error measures for Kirchhoff–Love cylindrical shells are validated by numerical results in Section 4, which is then followed by the conclusions in Section 5.

2 Isogeometric formulation for Kirchhoff–Love cylindrical shells

2.1 Governing equations of Kirchhoff–Love cylindrical shells

For a Kirchhoff–Love cylindrical shell as shown in Fig.1, the primary displacement variables associated with a general point \mathit{x} on the shell mid-surface \mathrm{\Omega } are the longitudinal displacement u, circumferential displacement v, and radial displacement or deflection w. Based upon the displacements, the membrane strain \epsilon and curvature \kappa of Kirchhoff–Love cylindrical shells are defined as:

(1)\epsilon (\mathit{x})=\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ 2{\epsilon }_{12}\end{array}\right\}=\left\{\begin{array}{c}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\\ \frac{1}{r}\frac{\mathrm{\partial }v}{\mathrm{\partial }\theta }+\frac{w}{r}\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }x}+\frac{1}{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }\theta }\end{array}\right\},

(2)\kappa (\mathit{x})=\left\{\begin{array}{c}{\kappa }_{11}\\ {\kappa }_{22}\\ 2{\kappa }_{12}\end{array}\right\}=\left\{\begin{array}{c}-\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{x}^2}\\ \frac{1}{r^2}\frac{\mathrm{\partial }v}{\mathrm{\partial }\theta }-\frac{1}{r^2}\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{\theta }^2}\\ \frac{1}{r}\frac{\mathrm{\partial }v}{\mathrm{\partial }x}-\frac{2}{r}\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }\theta \mathrm{\partial }x}\end{array}\right\},

where r is the shell radius.

The equations of motion for the cylindrical shell read [32]:

(3)C^{\mathrm{m}}\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{1-\nu }{2r^2}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\theta }^2}+\frac{\nu }{r}\frac{\mathrm{\partial }w}{\mathrm{\partial }x}+\frac{1+\nu }{2r}\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }x\mathrm{\partial }\theta }\right)=\rho \tau \ddot{u},

(4)\begin{array}{rl}& C^{\mathrm{m}}\left(\frac{1-\nu }{2}\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+\frac{1}{r^2}\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\theta }^2}+\frac{1}{r^2}\frac{\mathrm{\partial }w}{\mathrm{\partial }\theta }+\frac{1+\nu }{2r}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }\theta }\right)\\ +& C^{\mathrm{b}}\left(\frac{1-\nu }{2r^2}\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+\frac{1}{r^4}\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\theta }^2}-\frac{1}{r^4}\frac{{\mathrm{\partial }}^{3}w}{\mathrm{\partial }{\theta }^3}-\frac{1}{r^2}\frac{{\mathrm{\partial }}^{3}w}{\mathrm{\partial }{x}^2\mathrm{\partial }\theta }\right)=\rho \tau \ddot{v}\end{array},

(5)\begin{array}{rl}& C^{\mathrm{b}}\left(-\frac{{\mathrm{\partial }}^{4}w}{\mathrm{\partial }{x}^4}+\frac{1}{r^2}\frac{{\mathrm{\partial }}^{3}v}{\mathrm{\partial }{x}^2\mathrm{\partial }\theta }-\frac{2}{r^2}\frac{{\mathrm{\partial }}^{4}w}{\mathrm{\partial }{x}^2\mathrm{\partial }{\theta }^2}-\frac{1}{r^4}\frac{{\mathrm{\partial }}^{4}w}{\mathrm{\partial }{\theta }^4}+\frac{1}{r^4}\frac{{\mathrm{\partial }}^{3}v}{\mathrm{\partial }{\theta }^3}\right)\\ -& C^{\mathrm{m}}\left(\frac{1}{r^2}\frac{\mathrm{\partial }v}{\mathrm{\partial }\theta }+\frac{w}{r^2}+\frac{\nu }{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)=\rho \tau \ddot{w},\end{array}

where \rho is the material density, \tau is the shell thickness, C^{\mathrm{m}}=E\tau /(1-{\nu }^2), C^{\mathrm{b}}={\tau }^2{C}^{\mathrm{m}}/12, E and \nu are Young’s modulus and Poisson’s ratio, respectively. The weak form corresponding to Eqs. (3)–(5) can be stated as [26]:

(6){\int }_{\mathrm{\Omega }}^{}\delta \mathit{u}\rho \tau \ddot{\mathit{u}}\mathrm{d}\mathrm{\Omega }+{\int }_{\mathrm{\Omega }}^{}\delta {\epsilon }^{\mathrm{T}}{\mathit{C}}^{\mathrm{m}}\epsilon \mathrm{d}\mathrm{\Omega }+{\int }_{\mathrm{\Omega }}^{}\delta {\kappa }^{\mathrm{T}}{\mathit{C}}^{\mathrm{b}}\kappa \mathrm{d}\mathrm{\Omega }=0,

with

(7){\mathit{C}}^{\mathrm{m}}=\frac{E\tau }{1-{\nu }^2}\left[\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& (1-\nu )/2\end{array}\right],{\mathit{C}}^{\mathrm{b}}=\frac{{\tau }^2}{12}{\mathit{C}}^{\mathrm{m}}.

In the case of free vibration, the displacements can be assumed to take the following harmonic expressions [32]:

(8)\{\begin{array}{c}u(x,\theta ,t)=\overline{u}{\mathrm{e}}^{\iota (k_{x}x+k_{s}s-\omega t)},\\ v(x,\theta ,t)=\overline{v}{\mathrm{e}}^{\iota (k_{x}x+k_{s}s-\omega t)},\\ w(x,\theta ,t)=\overline{w}{\mathrm{e}}^{\iota (k_{x}x+k_{s}s-\omega t)},\end{array}

where \iota =\sqrt{-1}, s=r\theta. \overline{u}, \overline{v}, and \overline{w} are the wave amplitudes along the axial, circumferential, and radial directions, respectively. k_x and k_s are the longitudinal and circumferential wave numbers, respectively. \omega is the continuum frequency of the cylindrical shell. Bringing Eq. (8) into Eqs. (3)–(5) yields:

(9)\left[\begin{array}{ccc}{\omega }^2-d_{11}& d_{12}& d_{13}\\ d_{12}& {\omega }^2-d_{22}& d_{23}\\ -d_{13}& -d_{23}& {\omega }^2-d_{33}\end{array}\right]\left\{\begin{array}{c}\overline{u}\\ \overline{v}\\ \overline{w}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\},

where the coefficients d_{ij}^{\prime }s are given by:

(10)\{\begin{array}{c}{d}_{11}=c^2{k}_x^2+\frac{1-\nu }{2}{c}^2{k}_s^2,d_{12}=-\frac{1+\nu }{2}{c}^2{k}_x{k}_s,d_{13}=c^2\frac{\nu }{r}\iota k_x,\\ d_{22}=c^2{k}_s^2+\frac{1-\nu }{2}{c}^2{k}_x^2+\frac{1-\nu }{2r^2}{c}^2{\mu }^2{k}_x^2+c^2{\mu }^2\frac{1}{r^2}{k}_s^2,\\ d_{23}=\iota c^2{k}_s\frac{1}{r}+c^2{\mu }^2\frac{1}{r}\iota k_s^3+\frac{1}{r}{c}^2{\mu }^2\iota k_x^2{k}_s,\\ d_{33}=c^2{\mu }^2{k}_x^4+2c^2{\mu }^2{k}_x^2{k}_s^2+c^2{\mu }^2{k}_s^4+\frac{1}{r^2}{c}^2,\end{array}

in which c=\sqrt{E/[\rho (1-{\nu }^2)]} and \mu =\tau /\sqrt{12}.

The requirement of non-trivial solutions for Eq. (9) necessitates the determinant vanishing of the coefficient matrix, which then leads to the following characteristic equation:

(11)\begin{array}{rl}0=& -d_{33}{d}_{22}{d}_{11}-d_{11}{d}_{23}^2+d_{12}^2{d}_{33}-2d_{12}{d}_{13}{d}_{23}-d_{13}^2{d}_{22}\\ & +(d_{33}{d}_{11}+d_{22}{d}_{11}-d_{12}^2+d_{13}^2+d_{22}{d}_{33}+d_{23}^2){\omega }^2\\ & -(d_{11}+d_{22}+d_{33}){\omega }^4+{\omega }^6.\end{array}

Substituting Eq. (10) into Eq. (11) gives:

(12)\mathcal{F}(\omega )=\frac{1}{4r^4}(D_0{c}^6+D_2{\omega }^2{c}^4+D_4{\omega }^4{r}^2{c}^2+4{\omega }^6{r}^4)=0,

with

(13)\{\begin{array}{rl}& \begin{array}{rl}{D}_0=& {\nu }_{-1}^1\{2{\mu }^2{k}^8{r}^4+[2k_x^8{\mu }^4-k_s^2{\mu }^4{\nu }_{-5}^1{k}_x^6\\ & +2(-k_s^4{\nu }_{-2}^1{\mu }^4-2k_s^2{\nu }_{+2}^1{\mu }^2+1-{\nu }^2)k_x^4\\ & -k_s^4({\nu }_{-1}^1{k}_s^2{\mu }^2+{\nu }_{+12}^4){\mu }^2{k}_x^2-4k_s^6{\mu }^2]r^2\\ & +({\nu }_{+1}^{-1}{k}_x^2+2k_s^2)({\nu }_{+2}^2{k}_x^2+k_s^2){\mu }^2\},\end{array}\\ & \begin{array}{rl}{D}_2=& 2({\nu }_{+1}^{-1}{k}_x^2+2k_s^2){\mu }^2-2k^4(k_x^2{\nu }_{-3}^1{\mu }^2\\ & +k_s^2{\nu }_{-3}^1{\mu }^2+{\nu }_{-1}^1)r^4+[2{\mu }^4{\nu }_{-1}^1{k}_x^6\\ & +2{\mu }^2{\nu }_{-1}^1(2k_s^2{\mu }^2+1)k_x^4-(2k_s^4{\nu }_{-1}^1{\mu }^4\\ & -k_s^2{\nu }_{+1}^1{\nu }_{-3}^1{\mu }^2+4{\nu }^2+{\nu }_{-6}^2)k_x^2\\ & +2k_s^2(k_s^2{\nu }_{+3}^1{\mu }^2+{\nu }_{-1}^1)]r^2,\end{array}\\ & \begin{array}{rl}{D}_4=& -2k^2(2k_s^2{\mu }^2+2{\mu }^2{k_x}^2-{\nu }_{-3}^1)r^2\\ & +2{\mu }^2{k_x}^2({\nu }_{-1}^1-1)-4k_s^2{\mu }^2-4,\end{array}\end{array}

where {\nu }_{\pm n}^m=m\nu \pm n, k^2=k_x^2+k_s^2. As a result, it is noted that the analytical frequency \omega of the cylindrical shell satisfies the relationship of Eq. (12). Of course, proper boundary conditions are required to solve the frequency for specific problems. However, it is shown later that Eq. (12) is sufficient for the analytical frequency error investigation for cylindrical shell problems.

2.2 Isogeometric discretization

2.2.1 Basis functions

In the parametric space, the mid-surface \mathrm{\Omega } of a cylindrical shell is a rectangular domain, i.e., \mathrm{\Omega }(x,s)=[0,L]\times [0,r{\theta }_{\mathrm{T}}], where {\theta }_{\mathrm{T}}=2\text{π} for a complete cylindrical shell. By taking this advantage, we can directly employ the coordinates x and s to construct the isogeometric basis functions, say, the B-spline basis functions considered in this study. It is noted that a two-dimensional B-spline basis function can be conveniently formulated as the tensor product of two one-dimensional basis functions in each direction. Thus we start with the B-spline basis functions in the x direction, in this case, a B-spline basis function N_a^{[p]}(x) can be recursively defined as [2,33]:

(14)N_{\alpha }^{[p]}(x)=\{\begin{array}{ll}1,& x\in [x_a,x_{a+1}),\\ 0,& \text{otherwise},\end{array}\text{for}p=0,

(15)N_{\alpha }^{[p]}(x)=\frac{x-x_a}{x_{a+p}-x_a}{N}_a^{[p-1]}(x)+\frac{x_{a+p+1}-x}{x_{a+p+1}-x_{a+1}}{N}_{a+1}^{[p-1]}(x),\text{for}p⩾1,

where x\in [0,L], p denotes the basis degree, x_a stands for the ath knot of the following knot vector {\mathit{\vartheta }}_x:

(16){\mathit{\vartheta }}_x=\{x_1=0,...,x_{a-1},x_a,x_{a+1},...,x_{n_x+p+1}=L{\}}^{\mathrm{T}},

in which n_x represents the number of B-spline basis functions in the x direction. The knot intervals in {\mathit{\vartheta }}_x constitute the elements. Following a similar path, the B-spline basis functions in the circumferential direction N_b^{[p]}(s) can also be constructed in accordance with the knot vector {\mathit{\vartheta }}_s used in the circumferential direction, where the superscript b is used to denote the knot number in {\mathit{\vartheta }}_s.

For the convenience of subsequent development, the quadratic and cubic B-spline basis functions associated with a typical one dimensional element {\mathrm{\Omega }}_x^{\mathrm{e}}=[x_{a+1},x_{a+2}] are given as follows:

(17){\mathit{N}}_{\mathrm{e}}^{[2]}=\frac{1}{2h_{}^2}\left\{\begin{array}{c}(x_{a+2}-x{)}^2\\ (x-x_a)(x_{a+2}-x)+(x_{a+3}-x)(x-x_{a+1})\\ (x-x_{a+1}{)}^2\end{array}\right\},

(18){\mathit{N}}_{\mathrm{e}}^{[3]}=\frac{1}{6h_{}^3}\left\{\begin{array}{c}(x_{a+2}-x{)}^3\\ (x-x_{a-1})(x_{a+2}-x{)}^2+(x_{a+3}-x{)}^2(x-x_{a+1})\\ +(x_{a+3}-x)(x-x_a)(x_{a+2}-x)\\ (x-x_a{)}^2(x_{a+2}-x)+(x_{a+4}-x)(x-x_{a+1}{)}^2\\ +(x-x_a)(x_{a+3}-x)(x-x_{a+1})\\ (x-x_{a+1}{)}^3\end{array}\right\},

where h=x_{a+2}-x_{a+1} denotes the element length in the longitudinal direction. In the meantime, similar expressions hold for the basis functions in the circumferential direction.

It is noted that usually the open knot vectors whose first and last knots repeat (p+1) times are employed to construct B-spline basis functions. However, for a complete cylindrical shell, this basis construction approach is not well suitable for the circumferential discretization because such type of basis functions will lead to a non-physical C^0 approximation along the generatrix. To overcome this issue and construct a C^{p-1} approximation throughout the shell surface, similar to the enhanced meshfree approximation [34], a vector {\mathit{\vartheta }}_s with non-repeated knots and a periodicity of 2\text{π}r is used for the construction of N_b^{[p]}(s), and the conventional open knot vector {\mathit{\vartheta }}_x is employed to build up N_a^{[p]}(x). Accordingly, a product of N_a^{[p]}(x) and N_b^{[p]}(s) then constitutes the two-dimensional B-spline basis function with an enhancement of circumferential periodicity:

(19)N_{\mathrm{A}}^{[p]}(x,s)=N_{ab}^{[p]}(x,s)=N_a^{[p]}(x)N_b^{[p]}(s),

where the single subscript ‘A’ is used to represent the double subscript (a,b). The various quadratic B-spline basis functions are illustrated in Fig.2, where it is clear the proposed approach leads to C^1 smooth basis functions and the conventional basis functions exhibit a slope discontinuity along the shell generatrix.

The aforementioned B-spline basis functions are built upon the parametric space of the shell mid-surface. We also would like to remark that NURBS basis functions can be employed to exactly represent the cylindrical shell directly using the global Cartesian coordinates. However, since NURBS basis functions are rational, it is not convenient to employ them for an analytical frequency accuracy investigation.

2.2.2 Discrete equations

Based upon Eq. (19), the shell displacement field and related gradients are approximated as follows:

(20)\{\begin{array}{c}{\mathit{u}}^h(x,s)=\sum _{A=1}^{\mathcal{N}}{N}_A(x,s){\mathit{d}}_A,\\ {\mathit{u}}_{,x}^h(x,s)=\sum _{A=1}^{\mathcal{N}}{N}_{A,x}(x,s){\mathit{d}}_A,\\ {\mathit{u}}_{,s}^h(x,s)=\sum _{A=1}^{\mathcal{N}}{N}_{A,s}(x,s){\mathit{d}}_A,\end{array}

where {\mathit{u}}^{\mathrm{h}}=\{\begin{array}{ccc}{u}^{\mathrm{h}}& v^{\mathrm{h}}& w^{\mathrm{h}}\end{array}{\}}^{\mathrm{T}}, {\mathit{d}}_A=\{\begin{array}{ccc}{u}_A& v_A& w_A\end{array}{\}}^{\mathrm{T}}, and \mathcal{N} denotes the total number of basis functions. Accordingly, the discrete membrane strain and curvature vectors for the cylindrical shell read:

(21){\epsilon }^{\mathrm{h}}=\sum _{A=1}^{\mathcal{N}}{\mathit{B}}_A^{\mathrm{m}}{\mathit{d}}_A,{\kappa }^h=\sum _{A=1}^{\mathcal{N}}{\mathit{B}}_A^{\mathrm{b}}{\mathit{d}}_A,

with

(22){\mathit{B}}_A^{\mathrm{m}}=\left[\begin{array}{ccc}{N}_{A,x}& 0& 0\\ 0& N_{A,s}& {\displaystyle \frac{N_A}{r}}\\ N_{A,s}& N_{A,x}& 0\end{array}\right],{\mathit{B}}_A^{\mathrm{b}}=\left[\begin{array}{ccc}0& 0& -N_{A,xx}\\ 0& {\displaystyle \frac{N_{A,s}}{r}}& -N_{A,ss}\\ 0& {\displaystyle \frac{N_{A,x}}{r}}& -2N_{A,xs}\end{array}\right].

Substituting Eqs. (21) and (20) into the weak form of Eq. (6) leads to the isogeometric discrete equation of motion for cylindrical shells:

(23)\mathit{M}\ddot{\mathit{d}}+\mathit{K}\mathit{d}=\mathbf{0},

in which d stands for the global displacement coefficient vector, M and K are the global mass and stiffness matrices, which are constructed from their element counterparts:

(24)\mathit{M}=\underset{e=1}{\overset{n_{el}}{A}}[{\mathit{M}}^e\text{],}\mathit{K}=\underset{e=1}{\overset{n_{el}}{A}}[{\mathit{K}}^e\text{],}\mathit{d}=\underset{e=1}{\overset{n_{el}}{A}}[{\mathit{d}}^e\text{]},

where A is the local-global assembly operator. The element mass and stiffness matrices {\mathit{M}}^e and {\mathit{K}}^e are given by:

(25){\mathit{M}}^e=\left[\begin{array}{ccc}{\mathit{M}}_{\mathrm{u}\mathrm{u}}^e& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathit{M}}_{\mathrm{v}\mathrm{v}}^e& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathit{M}}_{\mathrm{w}\mathrm{w}}^e\end{array}\right],{\mathit{K}}^e=\left[\begin{array}{ccc}{\mathit{K}}_{\mathrm{u}\mathrm{u}}^e& {\mathit{K}}_{\mathrm{u}\mathrm{v}}^e& {\mathit{K}}_{\mathrm{u}\mathrm{w}}^e\\ {\mathit{K}}_{\mathrm{v}\mathrm{u}}^e& {\mathit{K}}_{\mathrm{v}\mathrm{v}}^e& {\mathit{K}}_{\mathrm{v}\mathrm{w}}^e\\ {\mathit{K}}_{\mathrm{w}\mathrm{u}}^e& {\mathit{K}}_{\mathrm{w}\mathrm{v}}^e& {\mathit{K}}_{\mathrm{w}\mathrm{w}}^e\end{array}\right],

with

(26){\mathit{M}}_{\mathrm{u}\mathrm{u}}^e={\mathit{M}}_{\mathrm{v}\mathrm{v}}^e={\mathit{M}}_{\mathrm{w}\mathrm{w}}^e={\int }_{{\mathrm{\Omega }}^e}^{}\rho \tau {\mathit{N}}^{e\mathrm{T}}{\mathit{N}}^e\mathrm{d}\mathrm{\Omega },

(27)\{\begin{array}{rl}& {\mathit{K}}_{\mathrm{u}\mathrm{u}}^e={\int }_{{\mathrm{\Omega }}^e}^{}{C}^{\mathrm{m}}({\mathit{N}}_{,x}^{e\mathrm{T}}{\mathit{N}}_{,x}^e-\frac{1-\nu }{2}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{,s}^e)\mathrm{d}\mathrm{\Omega },\\ & {\mathit{K}}_{\mathrm{u}\mathrm{v}}^e={\mathit{K}}_{\mathrm{v}\mathrm{u}}^{e\mathrm{T}}={\int }_{{\mathrm{\Omega }}^e}^{}{C}^{\mathrm{m}}(\nu {\mathit{N}}_{,x}^{e\mathrm{T}}{\mathit{N}}_{,s}^e+\frac{1-\nu }{2}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{,x}^e)\mathrm{d}\mathrm{\Omega },\\ & {\mathit{K}}_{\mathrm{u}\mathrm{w}}^e={\mathit{K}}_{\mathrm{w}\mathrm{u}}^{e\mathrm{T}}={\int }_{{\mathrm{\Omega }}^e}^{}{C}^{\mathrm{m}}\nu \frac{{\mathit{N}}_{,x}^{e\mathrm{T}}{\mathit{N}}_{}^e}{r}\mathrm{d}\mathrm{\Omega },\end{array}

(28)\{\begin{array}{rl}& \begin{array}{rl}{\mathit{K}}_{\mathrm{v}\mathrm{v}}^e=& {\int }_{{\mathrm{\Omega }}^e}^{}[C^{\mathrm{m}}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{,s}^e+\frac{1-\nu }{2}{C}^{\mathrm{m}}{\mathit{N}}_{,x}^{e\mathrm{T}}{\mathit{N}}_{,x}^e\\ & +\frac{1}{r^2}{C}^{\mathrm{b}}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{,s}^e+\frac{1-\nu }{2r^2}{C}^{\mathrm{b}}{\mathit{N}}_{,x}^{e\mathrm{T}}{\mathit{N}}_{,x}^e]\mathrm{d}\mathrm{\Omega },\end{array}\\ & \begin{array}{rl}{\mathit{K}}_{\mathrm{v}\mathrm{w}}^e=& {\mathit{K}}_{\mathrm{w}\mathrm{v}}^{e\mathrm{T}}=\frac{1}{r}{\int }_{{\mathrm{\Omega }}^e}^{}[C^{\mathrm{m}}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{}^e-\nu C^{\mathrm{b}}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{,xx}^e\\ & -C^{\mathrm{b}}{\mathit{N}}_{,s}^{e\mathrm{T}}{\mathit{N}}_{,ss}^e-(1-\nu )C^{\mathrm{b}}{\mathit{N}}_{,x}^{e\mathrm{T}}{\mathit{N}}_{,xs}^e]\mathrm{d}\mathrm{\Omega },\end{array}\\ & \begin{array}{rl}{\mathit{K}}_{\mathrm{w}\mathrm{w}}^e=& {\int }_{{\mathrm{\Omega }}^e}^{}[\frac{1}{r^2}{C}^{\mathrm{m}}{\mathit{N}}_{}^{e\mathrm{T}}{\mathit{N}}_{}^e+\nu C^{\mathrm{b}}{\mathit{N}}_{,ss}^{e\mathrm{T}}{\mathit{N}}_{,xx}^e+C^{\mathrm{b}}{\mathit{N}}_{,xx}^{e\mathrm{T}}{\mathit{N}}_{,xx}^e\\ & +\nu C^{\mathrm{b}}{\mathit{N}}_{,xx}^{e\mathrm{T}}{\mathit{N}}_{,ss}^e+C^{\mathrm{b}}{\mathit{N}}_{,ss}^{e\mathrm{T}}{\mathit{N}}_{,ss}^e+2(1-\nu )C^{\mathrm{b}}{\mathit{N}}_{,xs}^{e\mathrm{T}}{\mathit{N}}_{,xs}^e]\mathrm{d}\mathrm{\Omega }.\end{array}\end{array}

Subsequently, with the aid of the harmonic wave assumption, the discrete equation for the free vibration analysis of cylindrical shells is attained from Eq. (23) as [11]:

(29)\mathit{K}\varphi =({\omega }^{\mathrm{h}}{)}^2\mathit{M}\varphi ,

where {\omega }^{\mathrm{h}} is the isogeometric discrete frequency and \varphi is the corresponding vibration mode coefficient vector.

3 Frequency error analysis for isogeometric discretization of Kirchhoff–Love cylindrical shells

In this Section, under the circumstance of uniform mesh discretization that is characterized by a mesh length h_x in the longitudinal direction and width h_s along the circumferential direction, a theoretical frequency error analysis is presented for the isogeometric discretization of Kirchhoff–Love cylindrical shells using both quadratic and cubic basis functions. For brevity, we use “quadratic formulation” and “cubic formulation” to denote the isogeometric formulations with quadratic and cubic basis functions, respectively.

3.1 Frequency error analysis of quadratic formulation

To carry out the accuracy analysis, the harmonic wave expressions are assumed at the discrete level [11]:

(30){\mathit{d}}_A(t)={\mathit{d}}_{ab}(t)=\tilde{\mathit{u}}{\mathrm{e}}^{\iota (k_x{x}_a+k_s{s}_b-{\omega }^{\mathrm{h}}t)},

where the vector \stackrel{\mathbf{~}}{\mathit{u}}=\{\tilde{u}\tilde{v}\tilde{w}\} consists of the wave amplitudes in different directions, k_x and k_s are the wave numbers in axial and circumferential directions, x_a and s_b are the axial and circumferential coordinates of point A, respectively. In accordance with Eq. (30), there exist the following relationships:

(31){\ddot{\mathit{d}}}_A(t)={\ddot{\mathit{d}}}_{ab}(t)=-({\omega }^{\mathrm{h}}{)}^2\stackrel{\mathbf{~}}{\mathit{u}}{\mathrm{e}}^{\iota (k_x{x}_a+k_s{s}_b-{\omega }^{\mathrm{h}}t)},

(32){\mathit{d}}_{(a+m_x)(b+m_s)}(t)={\mathrm{e}}^{\iota (k_x{m}_x{h}_x+k_s{m}_s{h}_s)}{\mathit{d}}_{ab}(t),

where m_x and m_s are certain integers.

Based upon the explicit basis functions given by Eqs. (17) and (19), the element stiffness and mass matrices described by Eqs. (26)–(28) can be readily computed for uniform meshes. With these matrices in hand, through a very lengthy but straightforward derivation, the following relationship can be established via substituting Eqs. (30)–(32) into the three generic rows of Eq. (23):

(33)\left[\begin{array}{ccc}{c}^2{\mathcal{A}}_{11}-\mathcal{H}{\lambda }^h& c^2{\mathcal{A}}_{12}& \iota c^2{\mathcal{A}}_{13}\\ c^2{\mathcal{A}}_{12}& c^2{\mathcal{A}}_{22}-\mathcal{H}{\lambda }^h& \iota c^2{\mathcal{A}}_{23}\\ -\iota c^2{\mathcal{A}}_{13}& -\iota c^2{\mathcal{A}}_{23}& c^2{\mathcal{A}}_{33}-\mathcal{H}{\lambda }^h\end{array}\right]\left\{\begin{array}{c}\tilde{u}\\ \tilde{v}\\ \tilde{w}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\},

where {\lambda }^{\mathrm{h}}=({\omega }^{\mathrm{h}}{)}^2, the coefficients {\mathcal{A}}_{ij} and \mathcal{H} are listed in Electronic Supplementary Material as Eqs. (A1)–(A7). Consequently, the non-trivial solution requirement of Eq. (33) implies the determinant vanishing of the coefficient matrix, based on the Taylor expansion, we can obtain the following characteristic equation [20]:

(34){\mathcal{C}}_3^{}({\lambda }_{}^{\mathrm{h}}{)}_{}^3+{\mathcal{C}}_2^{}({\lambda }_{}^{\mathrm{h}}{)}_{}^2+{\mathcal{C}}_1^{}{\lambda }_{}^{\mathrm{h}}+{\mathcal{C}}_0^{}=0,

where {\mathcal{C}}_i are given in Electronic Supplementary Material. It is noted that Eq. (29) represents the whole system of equations for the generalized eigenvalue problem, and Eq. (34) contains the generic stencils arising from Eq. (29).

Further introducing the following frequency error measures:

(35)e=\frac{{\omega }^{\mathrm{h}}}{\omega }-1,\tilde{e}=\frac{{\lambda }^{\mathrm{h}}}{\lambda }-1,

where \lambda ={\omega }^2. Then we have e\approx \tilde{e}/2 and {\lambda }^{\mathrm{h}}=(1+\tilde{e})\lambda [35], and accordingly, Eq. (34) reduces to:

(36){\lambda }_{}^3{\mathcal{C}}_3^{}+{\lambda }_{}^2{\mathcal{C}}_2^{}+\lambda {\mathcal{C}}_1^{}+{\mathcal{C}}_0^{}+(3{\lambda }_{}^3{\mathcal{C}}_3^{}+2{\lambda }_{}^2{\mathcal{C}}_2^{}+\lambda {\mathcal{C}}_1^{})\tilde{e}\approx 0,

where the higher order terms of \tilde{e} are rationally dropped [20,35]. Finally, Eq. (36) gives the following error frequency measure for quadratic formulation:

(37)\begin{array}{rl}e\approx & \frac{1}{2}\tilde{e}\approx -\frac{{\lambda }_{}^3{\mathcal{C}}_3^{}+{\lambda }_{}^2{\mathcal{C}}_2^{}+\lambda {\mathcal{C}}_1^{}+{\mathcal{C}}_0^{}}{6{\lambda }_{}^3{\mathcal{C}}_3^{}+4{\lambda }_{}^2{\mathcal{C}}_2^{}+2\lambda {\mathcal{C}}_1^{}}\approx -\frac{1}{\mathcal{L}}\underset{=0}{\underbrace{\mathcal{F}(\omega )}}\\ & +\frac{1}{48\mathcal{L}{r}^4}(D_1^{[2]}{c}^6+D_2^{[2]}\lambda c^4+D_3^{[2]}{\lambda }^2{r}^2{c}^2+36k_{\alpha }^2{\lambda }^3{r}^4)h^2\\ \approx & \mathcal{O}(h^2),\end{array}

in which Eq. (12) is used. The coefficients \mathcal{L} and D_i^{[2]} are defined as follows:

(38)\begin{array}{rl}\mathcal{L}=& -\frac{1}{2r^4}\lambda \{2({\mu }^2{\nu }_{-3}^1{k}_x^2+k_s^2{\nu }_{-3}^1{\mu }^2+{\nu }_{-1}^1)k^4{r}^4\\ & +[2{\mu }^4{\nu }_{-1}^1{k}_x^6+2{\nu }_{-1}^1{\mu }^2(2k_s^2{\mu }^2+1)k_x^4\\ & +(2k_s^4{\nu }_{-1}^1{\mu }^4-k_s^2{\nu }_{+1}^1{\nu }_{-3}^1{\mu }^2+2\nu {\nu }_{+1}^2-6)k_x^2\\ & +2(k_s^2{\nu }_{+3}^1{\mu }^2+{\nu }_{-1}^1)k_s^2]r^2+{\mu }^2({\nu }_{-1}^1{k}_x^2-2k_s^2)\}{c}^4\\ & +2[(4k^2{\nu }^2-{\nu }_{-6}^2)k^2{r}^2-2{\nu }_{-1}^1{\mu }^2{k}_x^2\\ & +4k_s^2{\mu }^2+4]r^2{\lambda }^2{c}^2-12{\lambda }^3{r}^4,\end{array}

(39)\begin{array}{rl}{D}_1^{[2]}=& {\nu }_{-1}^1{\mu }^2{k}^4[16{\alpha }^2{k}_x^6+18k_s^2(2{\alpha }^2+1)k_x^4\\ & +18k_s^4({\alpha }^2+2)k_x^2+16k_s^6]r^4\\ & +\{16{\alpha }^2{\mu }^4{\nu }_{-1}^1{k}_x^{10}-2k_s^2[(2\nu {\nu }_{-13}^2+20){\alpha }^2-9{\nu }_{-1}^1]{\mu }^4{k}_x^8\\ & -[{\mu }^4{k}_s^4({\nu }_{-34}^{18}{\alpha }^2+{\nu }_{-45}^9)+36{\alpha }^2{\mu }^2{\nu }_{+2}^1{k}_s^2\\ & +18{\alpha }^2{\nu }_{-1}^1{\nu }_{+1}^1]{\nu }_{-1}^1{k}_x^6-[{\mu }^4{k}_s^4({\nu }_{-9}^9{\alpha }^2+{\nu }_{-34}^{18})\\ & +36{\mu }^2{k}_s^2({\alpha }^2{\nu }_{+3}^1+{\nu }_{+2}^1)+18{\nu }_{-1}^1{\nu }_{+1}^1]{\nu }_{-1}^1{k}_s^2{k}_x^4\\ & -{\mu }^2[(8{\nu }^2-2{\nu }_{-1}^4){\mu }^2{k}_s^2+36{\nu }_{-1}^1({\alpha }^2+{\nu }_{+3}^1)]k_s^6{k}_x^2\\ & -2k^8{\mu }^2(k_s^2{\mu }^2+18){\nu }_{-1}^1\}{r}^2\\ & +({\nu }_{+2}^2{k}_x^2+k_s^2)({\nu }_{+9}^{-9}{k}_x^2+18k_s^2)({\nu }_{-1}^1){\mu }^2{k}_{\alpha }^2,\end{array}

(40)\begin{array}{rl}{D}_2^{[2]}=& -\{16{\alpha }^2{\mu }^2{\nu }_{-3}^1{k}_x^6+18[(2{\alpha }^2+1){\nu }_{-3}^1{\mu }^2{k}_s^2\\ & +{\alpha }^2{\nu }_{-1}^1]k_x^4+[{\mu }^2({\alpha }^2+2){\nu }_{-3}^1{k}_s^2+({\alpha }^2+1){\nu }_{-1}^1]k_s^2{k}_x^2\\ & +[2(8k^2{\nu }_{-3}^1{\mu }^2+{\nu }_{-9}^9)]k_s^4\}{k}^2{r}^4\\ & +\{16{\alpha }^2{\mu }^4{\nu }_{-1}^1{k}_x^8+2[{\mu }^2({\nu }_{-16}^{18}{\alpha }^2+{\nu }_{-9}^9)k_s^2\\ & +9{\alpha }^2{\nu }_{-1}^1]{\mu }^2{k}_x^6+[18{\mu }^4({\alpha }^2+2){\nu }_{-1}^1{k}_s^4\\ & -9{\mu }^2({\alpha }^2{\nu }_{+1}^1{\nu }_{-3}^1-{\nu }_{-2}^2)k_s^2+18{\nu }_{-1}^1{\nu }_{+3}^2{\alpha }^2]k_x^4\\ & +[16{\mu }^4{\nu }_{-1}^1{k}_s^4+9({\nu }_{+6}^2{\alpha }^2-{\nu }_{+1}^1{\nu }_{-3}^1){\mu }^2{k}_s^2\\ & +{\nu }_{-1}^1({\alpha }^2+{\nu }_{+3}^2)]k_s^2{k}_x^2+[2(2k_s^4{\mu }^4+9k_s^2{\nu }_{+3}^1{\mu }^2\\ & +{\nu }_{-9}^9)]k_s^4\}{r}^2+(18{\nu }_{+1}^{-1}{k}_x^2+36k_s^2){\mu }^2{k}_{\alpha }^2,\end{array}

(41)\begin{array}{rl}{D}_3^{[2]}=& -\{32k_x^6{\alpha }^2{\mu }^2+18[2{\mu }^2(2{\alpha }^2+1)k_s^2-{\alpha }^2{\nu }_{-3}^1]k_x^4\\ & +18[2{\mu }^2({\alpha }^2+2)k_s^2-({\alpha }^2+1){\nu }_{-3}^1]k_s^2{k}_x^2\\ & +2(16k_s^2{\mu }^2-{\nu }_{-27}^9)k_s^4\}{r}^2\\ & +(36k_s^2{\mu }^2+36-18{\nu }_{-1}^1{\mu }^2{k}_x^2)k_{\alpha }^2,\end{array}

where h_x = αh_s = αh and k_{\alpha }^2={\alpha }^2{k}_x^2+k_s^2. From the error estimate described by Eq. (37), it is evident that the quadratic formulation produces the 2nd order accurate frequency results for the free vibration analysis of Kirchhoff–Love cylindrical shells.

3.2 Frequency error analysis of cubic formulation

Regarding the cubic formulation, the corresponding coefficients for the stencil equations in Eq. (33) are listed in Electronic Supplementary Material as Eqs. (A8)–(A23). Following a similar path as the previous quadratic formulation, the frequency error for cubic formulation can be attained as follows:

(42)\begin{array}{rl}e\approx & \frac{1}{2}\tilde{e}\approx \frac{(1-k_{\mathrm{a}}^2{h}^2)}{\mathcal{L}}\underset{=0}{\underbrace{\mathcal{F}(\omega )}}-\frac{1}{2880\mathcal{L}{r}^4}(D_1^{[3]}{c}^6\\ & +D_2^{[3]}\lambda c^4+D_3^{[3]}{\lambda }^2{r}^2{c}^2+D_4^{[3]}{r}^4{\lambda }^3)h^4\approx \mathcal{O}(h^4),\end{array}

with

(43)\begin{array}{rl}{D}_1^{[3]}=& 710{\alpha }^4{r}^2{\mu }^2({\mu }^2+r^2){\nu }_{-1}^1{k}_x^{12}+{\mu }^2{\alpha }^2{k}_s^2\{(2836{\alpha }^2+1440){\nu }_{-1}^1{r}^2\\ & -{\mu }^2[(355{\nu }^2+{\nu }_{+1775}^{-2126}){\alpha }^2-1440{\nu }_{-1}^1]\}{r}^2{k}_x^{10}\\ & +4250{\nu }_{-1}^1\{{\mu }^2[(4250{\alpha }^4+5760{\alpha }^2+708)r^2\\ & -{\mu }^2(-708+{\nu }_{-1418}^{708}{\alpha }^4+{\nu }_{-3600}^{720}{\alpha }^2)]r^2{k}_s^4\\ & -1416{\alpha }^4{r}^2{\mu }^2({\nu }^2+{\nu }_{+2}^1)k_s^2-708m^4({\nu }_{-1}^1{)}^2{\nu }_{+1}^1({\mu }^2+r^2)\}{k}_x^8\\ & +\{{\mu }^2[(2832{\alpha }^4+8640m^2+2832)r^2\\ & -{\mu }^2(354{\nu }_{-1}^1{\alpha }^4+1440{\nu }_{-2}^1{\alpha }^2+{\nu }_{-1770}^{354})]r^2{k}_s^4\\ & -24{\mu }^2{\alpha }^2(59{\nu }_{+3}^1{\alpha }^2+{\nu }_{+240}^{120})r^2{k}_s^2\\ & +{\alpha }^2[1440{\nu }_{+1}^{-1}{\nu }_{+1}^1{r}^2+6{\mu }^2({\nu }_{+295}^{177}{\alpha }^2-240{\nu }^2\\ & +240)]\}{k}_s^2{\nu }_{-1}^1{k}_x^6-\{{\mu }^2[(708{\alpha }^4+5760{\alpha }^2+4250)r^2\\ & -{\mu }^2(720{\nu }_{-1}^1{\alpha }^2+{\nu }_{-1418}^{708})]{\nu }_{-1}^1{r}^2{k}_s^4-2{\mu }^2[708{\nu }_{-1}^1{\alpha }^4\\ & +1440({\nu }^2+3){\alpha }^2+{\nu }_{+2}^1]r^2{k}_s^2+708{\nu }_{+1}^{-1}{\nu }_{+1}^1{r}^2\\ & +{\mu }^2(708{\alpha }^4+720{\nu }_{+5}^3{\alpha }^2+708{\nu }_{+1}^{-1}{\nu }_{+1}^1){\nu }_{-1}^1\}{k}_s^4{k}_x^4\\ & -k_s^6\{{\mu }^2[(1440{\alpha }^2+2836){\nu }_{-1}^1{r}^2-{\mu }^2(355{\nu }^2+{\nu }_{+359}^{-710})]r^2{k}_s^4\\ & -24{\mu }^2{\nu }_{-1}^1(120{\alpha }^2-59{\nu }^2-177)r^2{k}_s^2+6{\mu }^2{\nu }_{-1}^1(240{\alpha }^2\\ & +{\nu }_{+295}^{177})\}{k}_x^2-{\mu }^2[708+(710r^4+2r^2{\mu }^2)k_s^4\\ & -1416r^2{k}_s^2]k_s^8{\nu }_{-1}^1,\end{array}

(44)\begin{array}{rl}{D}_2^{[3]}=& 710{\mu }^2{\alpha }^4({\nu }_{-3}^1{r}^2+{\nu }_{-1}^1{\mu }^2)r^2{k}_x^{10}\\ & +2126{\alpha }^2{r}^2{k}_x^8\{{\mu }^2[{\nu }_{-3}^1(2126{\alpha }^2+1440)r^2\\ & +{\mu }^2({\nu }_{-1420}^{1416}{\alpha }^2+1440{\nu }_{-1}^1)]k_s^2+708{\alpha }^2({\mu }^2+r^2){\nu }_{-1}^1\}\\ & +\{{\mu }^2[3{\nu }_{-3}^1(708{\alpha }^4+1440{\alpha }^2+236)r^2\\ & +12{\mu }^2{\nu }_{-1}^1(59{\alpha }^4+240{\alpha }^2+59)]r^2{k}_s^4\\ & +2{\alpha }^2[12(59{\alpha }^2+60){\nu }_{-1}^1{r}^2-(177{\nu }_{+1}^1{\nu }_{-3}^1{\alpha }^2\\ & -720{\nu }_{-1}^1){\mu }^2]r^2{k}_s^2+708({\nu }_{+3}^2{r}^2+{\mu }^2){\alpha }^4{\nu }_{-1}^1\}{k}_x^6\\ & +\{{\mu }^2[12{\nu }_{-3}^1(59{\alpha }^4+360{\alpha }^2+177)r^2+{\mu }^2(1440{\alpha }^2\\ & +1416){\nu }_{-1}^1]r^2{k}_s^4+[12{\nu }_{-1}^1(59{\alpha }^4+240{\alpha }^2\\ & +59)r^2+{\mu }^2(708{\nu }_{+3}^1{\alpha }^4+708{\nu }_{-1}^1\\ & -720{\nu }_{+1}^1{\nu }_{-3}^1{\alpha }^2)]r^2{k}_s^2+12{\alpha }^2[(59{\alpha }^2+240\nu \\ & +360){\nu }_{-1}^1{r}^2-24{\mu }^2(-60{\nu }_{-1}^1+59{\alpha }^2)]\}{k}_s^2{k}_x^4\\ & +\{{\mu }^2[{\nu }_{-3}^1(1440{\alpha }^2+2126)r^2+710{\nu }_{-1}^1{\mu }^2]r^2{k}_s^4\\ & +[(1440{\alpha }^2+1416){\nu }_{-1}^1{r}^2+{\mu }^2(1440{\nu }_{+3}^1{\alpha }^2\\ & -354{\nu }_{+1}^1{\nu }_{-3}^1)]r^2{k}_s^2+24{\nu }_{-1}^1(60{\alpha }^2+59\nu \\ & +59)r^2-12{\mu }^2(240{\alpha }^2-59\nu +59)]\}{k}_s^4{k}_x^2\\ & +\{[{\mu }^2(710{\nu }_{-3}^1{r}^2-4{\mu }^2)r^2{k}_s^4\\ & +708r^2({\nu }_{-1}^1{r}^2+{\mu }^2{\nu }_{+3}^1)k_s^2+708{\nu }_{-1}^1{r}^2-1416{\mu }^2]\}{k}_s^6,\end{array}

(45)\begin{array}{rl}{D}_3^{[3]}=& 1420{\alpha }^4{r}^2{k}_x^8{\mu }^2+2{\alpha }^2[24r^2{\mu }^2(59{\alpha }^2+60)k_s^2\\ & -354{\alpha }^2({\nu }_{-3}^1{r}^2+{\nu }_{-1}^1{\mu }^2)]k_x^6\\ & +\{24r^2{\mu }^2(59{\alpha }^4+240{\alpha }^2+59)k^4\\ & -{\alpha }^2[12{\nu }_{-3}^1(59{\alpha }^2+120)r^2-24{\mu }^2(59{\alpha }^2\\ & -60{\nu }_{-1}^1)]k_s^2+1416{\alpha }^4\}{k}_x^4+\{[48r^2{\mu }^2(60{\alpha }^2+59)k_s^4\\ & +[12{\mu }^2(240{\alpha }^2-59\nu +59)-12{\nu }_{-3}^1(59{\alpha }^2\\ & +120)r^2]k_s^2+2880{\alpha }^2\}{k}_s^2{k}_x^2+1420r^2{k}_s^8{\mu }^2\\ & +1440({\nu }_{+3}^{-1}{r}^2+2{\mu }^2)k_s^6+2880k_s^4,\end{array}

(46)D_4^{[3]}=-24(59{\alpha }^4{k}_x^4+120{\alpha }^2{k}_s^2{k}_x^2+59k_s^4).

According to Eq. (42), it can be seen that a 4th order frequency accuracy is achieved by the isogeometric formulation for cylindrical Kirchhoff–Love shells using cubic basis functions.

4 Numerical demonstration

To systematically validate the proposed theoretical frequency error estimates in Section 3, the free vibration of a simply supported Kirchhoff–Love cylindrical shell is considered herein, whose geometry and material properties are: length L=30 m, radius r=10 m, thickness \tau =1 m, Young’s modulus E={10}^{\text{5}} MPa and Poisson’s ratio \nu =0.3. The analytical frequency solution for this shell problem is detailed in Electronic Supplementary Material, where m=k_{x}L/\text{π} and n=k_{s}r. During the frequency convergence study, as shown in Fig.3, three progressively refined meshes with 1125, 2000, and 3125 elements are used.

First, in order to illustrate the necessity for the employment of basis functions with enhanced circumferential periodicity, Fig.4 portrays the analytical mode corresponding to m=1 and n=2, and the numerical nodes generated by the quadratic isogeometric analysis using the standard open knot basis functions and the circumferential periodicity enhanced basis functions with 1125 elements. From Fig.4, it is obvious the periodicity enhanced basis functions successfully remove the slope discontinuity arising from the conventional open knot basis functions. Subsequently, these basis functions with circumferential periodicity are utilized to carry out the algorithmic convergence study, and the first 12 vibration modes are depicted in Fig.5 for the quadratic formulation. In Fig.6 and Fig.7, the convergence results for both quadratic and cubic formulations are presented, where e_i^{\mathrm{h}} and e_i denote the numerical error obtained by the direct isogeometric analysis and the theoretical error given by Eqs. (37) and (42) for the i\text{th} frequency. These numerical results congruously reveal that the theoretical convergence rates of 2 and 4 are well matched by the isogeometric frequency analysis of Kirchhoff–Love cylindrical shells with quadratic and cubic basis functions, respectively. This excellent agreement regarding frequency accuracy is further validated in Fig.8 and Fig.9 for the error maps produced by the numerical computation as well as the theoretical prediction.

5 Conclusions

An in-depth frequency error study was presented for the isogeometric free vibration analysis of the widely used Kirchhoff–Love cylindrical shells, where the quadratic and cubic basis functions were particularly considered. Owing to the complexity of shell formulation, rather than sticking to search for an analytical continuum frequency, the relationship that governs the continuum frequency was elaborated and then embedded into the accuracy analysis to facilitate the development of frequency error measures. Subsequently, with the aid of harmonic wave assumption for the shell displacement field, the stencil equations coupling the axial, circumferential and deflectional displacements were systematically derived for Kirchhoff–Love cylindrical shells. Accordingly, the frequency error measures were obtained for the isogeometric formulation with quadratic and cubic basis functions. It was proven that the theoretical frequency accuracy orders for the isogeometric analysis of Kirchhoff–Love cylindrical shells using quadratic and cubic basis functions are 2 and 4, respectively. These theoretical frequency error estimates were well demonstrated through their very close agreement with the numerical results regarding both accuracy orders and error magnitudes.

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