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A least squares recursive gradient meshfree collocation method for superconvergent structural vibration analysis

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Abstract

A least squares recursive gradient meshfree collocation method is proposed for the superconvergent computation of structural vibration frequencies. The proposed approach employs the recursive gradients of meshfree shape functions together with smoothed shape functions in the context of least squares formulation, where both meshfree nodes and auxiliary points are taken as the collocation points. It turns out that this least squares formulation can effectively suppress the spurious modes arising from a direct meshfree collocation formulation using recursive gradients. Meanwhile, a detailed theoretical analysis with explicit frequency error measure is presented for the least squares recursive gradient meshfree collocation method in order to assess the frequency accuracy of structural vibrations. This analysis discloses the salient basis degree discrepancy issue regarding the frequency accuracy for the least squares meshfree collocation formulation, and it is shown that this issue can be essentially resolved by the proposed least squares recursive gradient meshfree collocation method. In fact, the proposed method leads to superconvergent vibration frequencies when odd degree basis functions are used, i.e., the frequency convergence rate is improved from \((p - 1)\) for the standard least squares meshfree collocation to \((p + 1)\) for the proposed approach in case of an odd pth degree basis function. This desirable frequency superconvergence of the proposed least squares recursive gradient meshfree collocation method is congruously demonstrated by numerical results.

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Acknowledgements

The support of this work by the National Natural Science Foundation of China (12072302, 11772280) is gratefully acknowledged.

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  1. Department of Civil Engineering, Xiamen University, Xiamen, 361005, Fujian, China

    Like Deng, Dongdong Wang & Dongliang Qi

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  1. Like Deng
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Correspondence to Dongdong Wang.

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Appendix

Appendix

Equation Section (Next).

In this Appendix, the terms \({}^{{2N}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{xx}}\),\({}^{{2N}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{yy}}\),\({}^{{2N}}{\tilde{\mathcal{S}}}_{{(m,n)}}\), \({}^{{2N + 1}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{xx}}\) and \({}^{{2N + 1}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{yy}}\) in Eqs. (72)–(74) and (76)–(77) for the least squares recursive gradient meshfree collocation method, are detailed as follows:

$$ \begin{aligned} {}^{{2N}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{xx}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{2\delta _{{1m}} \delta _{{0n}} }}} } \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 2}} }}} + \cdots } \right\} \\ \approx & \underbrace {{\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } }}_{{ = {\mathcal{A}}}}k^{2} \cos ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \approx & {\mathcal{A}}k^{2} \cos ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \end{aligned} $$
(92)
$$ \begin{aligned} {}^{{2N}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{yy}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 2\delta _{{0m}} \delta _{{1n}} }}} \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 2}} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } k^{2} \sin ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \approx & {\mathcal{A}}k^{2} \sin ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \end{aligned} $$
(93)
$$ \begin{aligned} {}^{{2N}}{\tilde{\mathcal{S}}}_{{(m,n)}} = & \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{J}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{J}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = \delta _{{0m}} \delta _{{0n}} }}} \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{J}^{{}} ({\user2{x}}_{C} )\underbrace {{\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 2}} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{(\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \approx & {\mathcal{A}} + {\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \end{aligned} $$
(94)
$$ \begin{aligned} {}^{{2N + 1}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{xx}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 2\delta _{{1m}} \delta _{{0n}} }}} \right. \\ & + \sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 0}} \\ & \left. { + \sum\limits_{{m = 0}}^{{N + 2}} {\frac{{(k\iota )^{{2N + 4}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 4 - 2m}} }}{{(2m)!(2N + 4 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(2N + 4 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 4}} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } k^{2} \cos ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 4 - 2m}} }}{{(2m)!(2N + 4 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \approx & {\mathcal{A}}k^{2} \cos ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \end{aligned} $$
(95)
$$ \begin{aligned} {}^{{2N + 1}}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{yy}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 2\delta _{{0m}} \delta _{{1n}} }}} \right. \\ & + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 0}} \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 2}} {\frac{{(k\iota )^{{2N + 4}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 4 - 2m}} }}{{(2m)!(2N + 4 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(2N + 4 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 4}} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } k^{2} \sin ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 4 - 2m}} }}{{(2m)!(2N + 4 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \approx & {\mathcal{A}}k^{2} \sin ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \end{aligned} $$
(96)

Meanwhile, \({}^{1}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{xx}}\),\({}^{1}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{yy}}\) and \({}^{1}{\tilde{\mathcal{S}}}_{{(m,n)}}\) in Eqs. (79)–(81) are given by:

$$ \begin{aligned} {}^{1}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{xx}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{0} {\sum\limits_{{n = 0}}^{0} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 0}}} \right. \\ & + \;\sum\limits_{{m = 0}}^{1} {\frac{{(k\iota )^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2 - 2m}} }}{{(2m)!(2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = \delta _{{1m}} }} \\ & \left. { + \;\sum\limits_{{m = 0}}^{2} {\frac{{(k\iota )^{4} (\cos \theta )^{{2m}} (\sin \theta )^{{4 - 2m}} }}{{(2m)!(4 - 2m)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\tilde{\Psi }_{{J,xx}}^{{[2]}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(4 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{4} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } k^{2} \cos ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{4 - 2m}} }}{{(2m)!(4 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{2} ] \\ \approx & {\mathcal{A}}k^{2} \cos ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{2} ] \\ \end{aligned} $$
(97)
$$ \begin{aligned} {}^{1}{\tilde{\mathcal{G}}}_{{(m,n)}}^{{yy}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{0} {\sum\limits_{{n = 0}}^{0} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 0}}} \right. \\ & + \;\sum\limits_{{m = 0}}^{1} {\frac{{(k\iota )^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2 - 2m}} }}{{(2m)!(2 - 2m)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )\psi _{{(2m)(2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = \delta _{{0m}} }} \\ & \left. { + \;\sum\limits_{{m = 0}}^{2} {\frac{{(k\iota )^{4} (\cos \theta )^{{2m}} (\sin \theta )^{{4 - 2m}} }}{{(2m)!(4 - 2m)!}}\sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\tilde{\Psi }_{{J,yy}}^{{[2]}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(4 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{4} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } k^{2} \sin ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{4 - 2m}} }}{{(2m)!(4 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{2} ] \\ \approx & {\mathcal{A}}k^{2} \sin ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{2} ] \\ \end{aligned} $$
(98)
$$ \begin{aligned} {}^{1}{\tilde{\mathcal{S}}}_{{(m,n)}}^{{}} = & \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\beta _{C} \tilde{\Psi }_{J}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & \left\{ {\sum\limits_{{m = 0}}^{0} {\sum\limits_{{n = 0}}^{0} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{J}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = \delta _{{0m}} \delta _{{0n}} }}} \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{1} {\frac{{(k\iota )^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2 - 2m}} }}{{(2m)!(2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \sum\limits_{{J = 1}}^{{NP}} {\tilde{\Psi }_{J}^{{}} ({\user2{x}}_{C} )\underbrace {{\psi _{{(2m)(2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{2} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } + \sum\limits_{{C = 1}}^{{NC}} {\beta _{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2 - 2m}} }}{{(2m)!(2 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{2} ] \\ \approx & {\mathcal{A}} + k^{2} {\mathcal{O}}[(k\alpha h)^{2} ] \\ \end{aligned} $$
(99)

On the other hand, for the standard least squares meshfree collocation method, the terms \({}^{{2N + 1}}{\bar{\mathcal{G}}}_{{(m,n)}}^{{xx}}\),\({}^{{2N + 1}}{\bar{\mathcal{G}}}_{{(m,n)}}^{{yy}}\) and \({}^{{2N + 1}}{\bar{\mathcal{S}}}_{{(m,n)}}\) in Eqs. (84)–(86) read:

$$ \begin{aligned} {}^{{2N + 1}}{\bar{\mathcal{G}}}_{{(m,n)}}^{{xx}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\Psi _{J}^{{}} ({\user2{x}}_{C} )}}_{{ = \bar{\beta }_{C} }}\Psi _{{J,xx}}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\Psi _{{J,xx}}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 2\delta _{{1m}} \delta _{{0n}} }}} \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\Psi _{{J,xx}}^{{}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 2}} }}} + \cdots } \right\} \\ \approx & \underbrace {{\sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } }}_{{ = {\bar{\mathcal{A}}}}}k^{2} \cos ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \approx & {\bar{\mathcal{A}}}k^{2} \cos ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \end{aligned} $$
(100)
$$ \begin{aligned} {}^{{2N + 1}}{\bar{\mathcal{G}}}_{{(m,n)}}^{{yy}} = & - \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\bar{\beta }_{C} \Psi _{{J,yy}}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & - \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\Psi _{{J,yy}}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = 2\delta _{{0m}} \delta _{{1n}} }}} \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \sum\limits_{{J = 1}}^{{NP}} {\underbrace {{\Psi _{{J,yy}}^{{}} ({\user2{x}}_{C} )}}_{{\sim 1/(\alpha h)^{2} }}\underbrace {{\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 2}} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } k^{2} \sin ^{2} \theta + \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \frac{{k^{2} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 4 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \approx & {\bar{\mathcal{A}}}k^{2} \sin ^{2} \theta + k^{2} {\mathcal{O}}[(k\alpha h)^{{2N}} ] \\ \end{aligned} $$
(101)
$$ \begin{aligned} {}^{{2N + 1}}{\bar{\mathcal{S}}}_{{(m,n)}} = & \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{n = 0}}^{\infty } {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\sum\limits_{{J = 1}}^{{NP}} {\bar{\beta }_{C} \Psi _{J}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} } \\ = & \left\{ {\sum\limits_{{m = 0}}^{N} {\sum\limits_{{n = 0}}^{{N - m}} {\frac{{(\iota k)^{{2m + 2n}} (\cos \theta )^{{2m}} (\sin \theta )^{{2n}} }}{{(2m)!(2n)!}}} } \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \underbrace {{\sum\limits_{{J = 1}}^{{NP}} {\Psi _{J}^{{}} ({\user2{x}}_{C} )\psi _{{(2m)(2n)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )} }}_{{ = \delta _{{0m}} \delta _{{0n}} }}} \right. \\ & \left. { + \;\sum\limits_{{m = 0}}^{{N + 1}} {\frac{{(k\iota )^{{2N + 2}} (\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}} \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \sum\limits_{{J = 1}}^{{NP}} {\Psi _{J}^{{}} ({\user2{x}}_{C} )\underbrace {{\psi _{{(2m)(2N + 2 - 2m)}} ({\user2{x}}_{J} - {\user2{x}}_{C} )}}_{{\sim (\alpha h)^{{2N + 2}} }}} + \cdots } \right\} \\ \approx & \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } + \sum\limits_{{C = 1}}^{{NC}} {\bar{\beta }_{C} } \frac{{(\cos \theta )^{{2m}} (\sin \theta )^{{2N + 2 - 2m}} }}{{(2m)!(2N + 2 - 2m)!}}{\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \approx & {\bar{\mathcal{A}}} + {\mathcal{O}}[(k\alpha h)^{{2N + 2}} ] \\ \end{aligned} $$
(102)

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Deng, L., Wang, D. & Qi, D. A least squares recursive gradient meshfree collocation method for superconvergent structural vibration analysis. Comput Mech 68, 1063–1096 (2021). https://doi.org/10.1007/s00466-021-02059-5

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